From 424f19fd3c6ed37863fc6ccabbc8fd50d2e9f64d Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 09:37:04 -0500 Subject: [PATCH 01/22] Inserted notes --- .../manipulating-derivatives.md | 37 +++++++++++++++++++ 1 file changed, 37 insertions(+) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index 9758b447..16e9c1ea 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -1 +1,38 @@ # Manipulating partial derivatives + +We sometimes need to find derivatives we don't obtain easily from one of these potentials. For this, we use some calculus rules: + +####Inversion: + +If x(y,z) and y(x,z) then + +{\frac{\partial x}{\partial y}}_z = \frac{1}{{\frac{partial y}{partial x}}_z} + +````{example} +x = \frac{y^2}{z} + +We can rearrange this to form y = \pm \sqrt{xz} + +{\frac{\partial x}{\partial y}}_z = \frac{2y}{z} + +{\frac{\partial y}{\partial x}}_z = {\frac{1}{2}}{\sqrt{\fract{2}{x}}} = {\frac{1}{2}}{\sqrt{\fract{z}{\frac{y^2}{z}}}} = \frac{x}{2y} + +\frac{1}{{partial y}{partial x}_z} = \frac{2y}{z} +```` + +####Chain Rule: + +If x(y,z) then + +{\frac{\partial x}{\partial y}}_z = {{\frac{\partial x}{\partial w}}_z}{{\frac{partial w}{\partial y}}_z} + +where {\frac{\partial x}{\partial w}}_z is x(w,z) and {\frac{partial w}{\partial y}}_z is w(y,z) + +````{example} +Define w = yz so y = w/z + +x = \frac{{w/z}^2}{z} = \frac{w^2}{z^3} + +{{\frac{\partial x}{\partial w}}_z} = \frac{2w}{z^3} and {\frac{partial w}{\partial y}}_z = z + +{{\frac{\partial x}{\partial w}}_z}{\frac{partial w}{\partial y}}_z = {\frac{2w}{z^3}}z = \frac{2w}{z^2} = \frac{2yz}{z^2} = \frac{2y}{z} From 5b1aabf776319da0a742add7f46d3fa3a8a9b787 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 09:51:41 -0500 Subject: [PATCH 02/22] issue-30 --- multivariable-calculus/manipulating-derivatives.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index 16e9c1ea..ee3e1511 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -2,7 +2,7 @@ We sometimes need to find derivatives we don't obtain easily from one of these potentials. For this, we use some calculus rules: -####Inversion: +#### Inversion If x(y,z) and y(x,z) then @@ -20,7 +20,7 @@ We can rearrange this to form y = \pm \sqrt{xz} \frac{1}{{partial y}{partial x}_z} = \frac{2y}{z} ```` -####Chain Rule: +#### Chain Rule If x(y,z) then From 09b5bdc46530e60c227f4be31bcdfe165e6a9c5d Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 09:58:29 -0500 Subject: [PATCH 03/22] issue-30 --- .../manipulating-derivatives.md | 72 +++++++++++-------- 1 file changed, 44 insertions(+), 28 deletions(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index ee3e1511..dc5c7b3c 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -4,35 +4,51 @@ We sometimes need to find derivatives we don't obtain easily from one of these p #### Inversion -If x(y,z) and y(x,z) then +If \( x(y, z) \) and \( y(x, z) \), then -{\frac{\partial x}{\partial y}}_z = \frac{1}{{\frac{partial y}{partial x}}_z} +$$ +\left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} +$$ ````{example} +Let +$$ x = \frac{y^2}{z} - -We can rearrange this to form y = \pm \sqrt{xz} - -{\frac{\partial x}{\partial y}}_z = \frac{2y}{z} - -{\frac{\partial y}{\partial x}}_z = {\frac{1}{2}}{\sqrt{\fract{2}{x}}} = {\frac{1}{2}}{\sqrt{\fract{z}{\frac{y^2}{z}}}} = \frac{x}{2y} - -\frac{1}{{partial y}{partial x}_z} = \frac{2y}{z} -```` - -#### Chain Rule - -If x(y,z) then - -{\frac{\partial x}{\partial y}}_z = {{\frac{\partial x}{\partial w}}_z}{{\frac{partial w}{\partial y}}_z} - -where {\frac{\partial x}{\partial w}}_z is x(w,z) and {\frac{partial w}{\partial y}}_z is w(y,z) - -````{example} -Define w = yz so y = w/z - -x = \frac{{w/z}^2}{z} = \frac{w^2}{z^3} - -{{\frac{\partial x}{\partial w}}_z} = \frac{2w}{z^3} and {\frac{partial w}{\partial y}}_z = z - -{{\frac{\partial x}{\partial w}}_z}{\frac{partial w}{\partial y}}_z = {\frac{2w}{z^3}}z = \frac{2w}{z^2} = \frac{2yz}{z^2} = \frac{2y}{z} +$$ + +We can rearrange this to form: +$$ +y = \pm \sqrt{xz} +$$ + +Then: +$$ +\left( \frac{\partial x}{\partial y} \right)_z = \frac{2y}{z} +$$ + +And: +$$ +\left( \frac{\partial y}{\partial x} \right)_z = \frac{1}{2} \cdot \frac{z}{\sqrt{xz}} = \frac{z}{2y} +$$ + +Thus: +$$ +\frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} = \frac{2y}{z} +$$ + +Define \( w = yz \), so \( y = \frac{w}{z} \) + +Then: +$$ +x = \frac{(w/z)^2}{z} = \frac{w^2}{z^3} +$$ + +Now compute: +$$ +\left( \frac{\partial x}{\partial w} \right)_z = \frac{2w}{z^3}, \quad \left( \frac{\partial w}{\partial y} \right)_z = z +$$ + +Therefore: +$$ +\left( \frac{\partial x}{\partial y} \right)_z = \left( \frac{2w}{z^3} \right) z = \frac{2w}{z^2} = \frac{2yz}{z^2} = \frac{2y}{z} +$$ From 490ae9b72cc92faba9a21c5f24dac7ebb500cba5 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 10:00:41 -0500 Subject: [PATCH 04/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index dc5c7b3c..87a99701 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -6,7 +6,7 @@ We sometimes need to find derivatives we don't obtain easily from one of these p If \( x(y, z) \) and \( y(x, z) \), then -$$ + \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} $$ From cb2ade47745f935d63b6fa6ed43098bb02c62480 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 10:02:15 -0500 Subject: [PATCH 05/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index 87a99701..063957bf 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -4,7 +4,7 @@ We sometimes need to find derivatives we don't obtain easily from one of these p #### Inversion -If \( x(y, z) \) and \( y(x, z) \), then +If \ x(y, z) \ and \ y(x, z) \, then \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} From d7827321d215f769c5d912d80d8e7e35fdba4db1 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 10:02:49 -0500 Subject: [PATCH 06/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index 063957bf..c54a7ba1 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -6,7 +6,7 @@ We sometimes need to find derivatives we don't obtain easily from one of these p If \ x(y, z) \ and \ y(x, z) \, then - +$$ \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} $$ From 44414822ace7498677a3f06fc64017b6739ab3f7 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 10:03:49 -0500 Subject: [PATCH 07/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index c54a7ba1..a602d0d4 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -4,7 +4,7 @@ We sometimes need to find derivatives we don't obtain easily from one of these p #### Inversion -If \ x(y, z) \ and \ y(x, z) \, then +If x(y, z) and y(x, z), then $$ \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} From 2253d1634e3778fabd009a3fe2b9ad7bd82787bd Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 10:08:42 -0500 Subject: [PATCH 08/22] issue30 --- .../manipulating-derivatives.md | 66 +++++++++---------- 1 file changed, 30 insertions(+), 36 deletions(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index a602d0d4..a05f9a26 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -1,54 +1,48 @@ -# Manipulating partial derivatives +# Manipulating Partial Derivatives We sometimes need to find derivatives we don't obtain easily from one of these potentials. For this, we use some calculus rules: #### Inversion -If x(y, z) and y(x, z), then +If \( x(y, z) \) and \( y(x, z) \), then: -$$ +```{math} \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} -$$ -````{example} -Let -$$ +Let: +```{math} x = \frac{y^2}{z} -$$ +``` -We can rearrange this to form: -$$ +We can rearrange this to: +```{math} y = \pm \sqrt{xz} -$$ +``` -Then: -$$ +Compute: +```{math} \left( \frac{\partial x}{\partial y} \right)_z = \frac{2y}{z} -$$ +``` -And: -$$ -\left( \frac{\partial y}{\partial x} \right)_z = \frac{1}{2} \cdot \frac{z}{\sqrt{xz}} = \frac{z}{2y} -$$ +Then: +```{math} +\left( \frac{\partial y}{\partial x} \right)_z = \frac{z}{2y} +``` -Thus: -$$ +And: +```{math} \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} = \frac{2y}{z} -$$ +``` -Define \( w = yz \), so \( y = \frac{w}{z} \) +\left( \frac{\partial x}{\partial y} \right)_z = \left( \frac{\partial x}{\partial w} \right)_z \left( \frac{\partial w}{\partial y} \right)_z -Then: -$$ -x = \frac{(w/z)^2}{z} = \frac{w^2}{z^3} -$$ - -Now compute: -$$ -\left( \frac{\partial x}{\partial w} \right)_z = \frac{2w}{z^3}, \quad \left( \frac{\partial w}{\partial y} \right)_z = z -$$ - -Therefore: -$$ -\left( \frac{\partial x}{\partial y} \right)_z = \left( \frac{2w}{z^3} \right) z = \frac{2w}{z^2} = \frac{2yz}{z^2} = \frac{2y}{z} -$$ +# Manipulating Partial Derivatives + +We sometimes need to find derivatives we don't obtain easily from one of these potentials. For this, we use some calculus rules: + +#### Inversion + +If \( x(y, z) \) and \( y(x, z) \), then: + +```{math} +\left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} From b8a97b222d2cd699c3f71d491b603b899bda9801 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 10:10:37 -0500 Subject: [PATCH 09/22] issue 30 --- .../manipulating-derivatives.md | 22 ++++++++++++++----- 1 file changed, 17 insertions(+), 5 deletions(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index a05f9a26..09fda327 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -33,16 +33,28 @@ And: ```{math} \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} = \frac{2y}{z} ``` +#### Chain Rule \left( \frac{\partial x}{\partial y} \right)_z = \left( \frac{\partial x}{\partial w} \right)_z \left( \frac{\partial w}{\partial y} \right)_z -# Manipulating Partial Derivatives +Define \( w = yz \), so \( y = \frac{w}{z} \) -We sometimes need to find derivatives we don't obtain easily from one of these potentials. For this, we use some calculus rules: +Then: +```{math} +x = \frac{(w/z)^2}{z} = \frac{w^2}{z^3} +``` -#### Inversion +Now compute: +```{math} +\left( \frac{\partial x}{\partial w} \right)_z = \frac{2w}{z^3} +``` -If \( x(y, z) \) and \( y(x, z) \), then: +```{math} +\left( \frac{\partial w}{\partial y} \right)_z = z +``` +Therefore: ```{math} -\left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} +\left( \frac{\partial x}{\partial y} \right)_z = \frac{2w}{z^3} \cdot z = \frac{2w}{z^2} = \frac{2yz}{z^2} = \frac{2y}{z} +``` + From d15b4fd2f8e085639d1c806b23747307a2cd4693 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 10:15:25 -0500 Subject: [PATCH 10/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index 09fda327..ea1b2742 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -10,7 +10,7 @@ If \( x(y, z) \) and \( y(x, z) \), then: \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} Let: -```{math} + x = \frac{y^2}{z} ``` From cd423c22ca78302e49d28a106c3fbb3052c8989f Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 10:18:33 -0500 Subject: [PATCH 11/22] issue30 --- .../manipulating-derivatives.md | 19 +++++++++++++------ 1 file changed, 13 insertions(+), 6 deletions(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index ea1b2742..dbfbff90 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -1,16 +1,20 @@ # Manipulating Partial Derivatives -We sometimes need to find derivatives we don't obtain easily from one of these potentials. For this, we use some calculus rules: +We sometimes need to find derivatives we don't obtain easily from one of these potentials. For this, we use some calculus rules. -#### Inversion +--- + +## Inversion If \( x(y, z) \) and \( y(x, z) \), then: ```{math} \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} -Let: +**Example:** +Let: +```{math} x = \frac{y^2}{z} ``` @@ -19,7 +23,7 @@ We can rearrange this to: y = \pm \sqrt{xz} ``` -Compute: +Now compute: ```{math} \left( \frac{\partial x}{\partial y} \right)_z = \frac{2y}{z} ``` @@ -29,14 +33,17 @@ Then: \left( \frac{\partial y}{\partial x} \right)_z = \frac{z}{2y} ``` -And: +And confirm the inversion: ```{math} \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} = \frac{2y}{z} ``` + #### Chain Rule \left( \frac{\partial x}{\partial y} \right)_z = \left( \frac{\partial x}{\partial w} \right)_z \left( \frac{\partial w}{\partial y} \right)_z +**Example:** + Define \( w = yz \), so \( y = \frac{w}{z} \) Then: @@ -49,6 +56,7 @@ Now compute: \left( \frac{\partial x}{\partial w} \right)_z = \frac{2w}{z^3} ``` +And: ```{math} \left( \frac{\partial w}{\partial y} \right)_z = z ``` @@ -57,4 +65,3 @@ Therefore: ```{math} \left( \frac{\partial x}{\partial y} \right)_z = \frac{2w}{z^3} \cdot z = \frac{2w}{z^2} = \frac{2yz}{z^2} = \frac{2y}{z} ``` - From 1a231110a039ddf15a1c66d43fdc0113596a9714 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 10:43:58 -0500 Subject: [PATCH 12/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index dbfbff90..280a6c38 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -40,8 +40,9 @@ And confirm the inversion: #### Chain Rule +```{msth} \left( \frac{\partial x}{\partial y} \right)_z = \left( \frac{\partial x}{\partial w} \right)_z \left( \frac{\partial w}{\partial y} \right)_z - +``` **Example:** Define \( w = yz \), so \( y = \frac{w}{z} \) From 3b47e6a2725302ea3fc2ab2047a2e9e6e4308ee0 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 12:06:08 -0500 Subject: [PATCH 13/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index 280a6c38..00cfd647 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -4,9 +4,9 @@ We sometimes need to find derivatives we don't obtain easily from one of these p --- -## Inversion +#### Inversion -If \( x(y, z) \) and \( y(x, z) \), then: +If x(y, z) and y(x, z), then: ```{math} \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} From 153a34380ba9c8f6aad603f7218664b1d0ccf50f Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 12:41:23 -0500 Subject: [PATCH 14/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index 00cfd647..12f51f7d 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -40,7 +40,7 @@ And confirm the inversion: #### Chain Rule -```{msth} +```{math} \left( \frac{\partial x}{\partial y} \right)_z = \left( \frac{\partial x}{\partial w} \right)_z \left( \frac{\partial w}{\partial y} \right)_z ``` **Example:** From ff7d607b3a5b4595ac613294eb895c72d9c4b7ce Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 12:42:19 -0500 Subject: [PATCH 15/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 1 - 1 file changed, 1 deletion(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index 12f51f7d..78783b03 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -14,7 +14,6 @@ If x(y, z) and y(x, z), then: **Example:** Let: -```{math} x = \frac{y^2}{z} ``` From baf7f88e3841774a24787830fa1f120d690a3c17 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 12:50:44 -0500 Subject: [PATCH 16/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index 78783b03..5d5ebe69 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -44,7 +44,7 @@ And confirm the inversion: ``` **Example:** -Define \( w = yz \), so \( y = \frac{w}{z} \) +Define w = yz , so y = \frac{w}{z} Then: ```{math} From 098d80178196674535d13fd4fbf540cbbfb42471 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 12:54:22 -0500 Subject: [PATCH 17/22] issue-30 --- multivariable-calculus/manipulating-derivatives.md | 1 + 1 file changed, 1 insertion(+) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index 5d5ebe69..ef7a1327 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -10,6 +10,7 @@ If x(y, z) and y(x, z), then: ```{math} \left( \frac{\partial x}{\partial y} \right)_z = \frac{1}{\left( \frac{\partial y}{\partial x} \right)_z} +``` **Example:** From cc339dcf841c02108bb3e075a19d9fb07e4b2cde Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 12:56:07 -0500 Subject: [PATCH 18/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index ef7a1327..dfe9540f 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -45,8 +45,9 @@ And confirm the inversion: ``` **Example:** -Define w = yz , so y = \frac{w}{z} - +```{math} + w = yz , so y = \frac{w}{z} +``` Then: ```{math} x = \frac{(w/z)^2}{z} = \frac{w^2}{z^3} From 8f133e9aab7660e27e1b56f660d903f787e83363 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 12:56:47 -0500 Subject: [PATCH 19/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 1 + 1 file changed, 1 insertion(+) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index dfe9540f..dc9865c1 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -14,6 +14,7 @@ If x(y, z) and y(x, z), then: **Example:** +```{math} Let: x = \frac{y^2}{z} ``` From 897b1b69946ab14c3729d53401f4d3f7390eaa9f Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 12:58:44 -0500 Subject: [PATCH 20/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index dc9865c1..4e5b5e1d 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -47,7 +47,7 @@ And confirm the inversion: **Example:** ```{math} - w = yz , so y = \frac{w}{z} + w = yz , so y = \frac{w}{z} ``` Then: ```{math} From 3cbe127815a083be9af35787161198b5ae3bc2a0 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 12:59:33 -0500 Subject: [PATCH 21/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 6 +++++- 1 file changed, 5 insertions(+), 1 deletion(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index 4e5b5e1d..2935255e 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -47,8 +47,12 @@ And confirm the inversion: **Example:** ```{math} - w = yz , so y = \frac{w}{z} + w = yz , ``` + so + ```{math} + y = \frac{w}{z} + ``` Then: ```{math} x = \frac{(w/z)^2}{z} = \frac{w^2}{z^3} From 37201795fd34bf844df8ff3d5002a0e9ea28fc88 Mon Sep 17 00:00:00 2001 From: mollygracehicks Date: Wed, 16 Apr 2025 13:01:19 -0500 Subject: [PATCH 22/22] issue30 --- multivariable-calculus/manipulating-derivatives.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/multivariable-calculus/manipulating-derivatives.md b/multivariable-calculus/manipulating-derivatives.md index 2935255e..f9fa4423 100644 --- a/multivariable-calculus/manipulating-derivatives.md +++ b/multivariable-calculus/manipulating-derivatives.md @@ -47,9 +47,9 @@ And confirm the inversion: **Example:** ```{math} - w = yz , + w = yz ``` - so + So, ```{math} y = \frac{w}{z} ```