the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function

Calculus Volume 1 · 3.6 The Chain Rule

The Chain Rule: practice problems

23 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 215
For the following exercises, given $y = f (u)$ and $u = g (x),$ find $\frac{d y}{d x}$ by using Leibniz’s notation for the chain rule: $\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x} .$
$y = 6 u^{3}, u = 7 x - 4$
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Exercises 217
For the following exercises, given $y = f (u)$ and $u = g (x),$ find $\frac{d y}{d x}$ by using Leibniz’s notation for the chain rule: $\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x} .$
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Exercises 219
For the following exercises, given $y = f (u)$ and $u = g (x),$ find $\frac{d y}{d x}$ by using Leibniz’s notation for the chain rule: $\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x} .$
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Exercises 221
For each of the following exercises,
decompose each function in the form $y = f (u)$ and $u = g (x),$ and
find $\frac{d y}{d x}$ as a function of $x .$
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Exercises 223
For each of the following exercises,
decompose each function in the form $y = f (u)$ and $u = g (x),$ and
find $\frac{d y}{d x}$ as a function of $x .$
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Exercises 225
For each of the following exercises,
decompose each function in the form $y = f (u)$ and $u = g (x),$ and
find $\frac{d y}{d x}$ as a function of $x .$
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Exercises 227
For each of the following exercises,
decompose each function in the form $y = f (u)$ and $u = g (x),$ and
find $\frac{d y}{d x}$ as a function of $x .$
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Exercises 229
For the following exercises, find $\frac{d y}{d x}$ for each function.
$y = (5 - 2 x)^{- 2}$
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Exercises 231
For the following exercises, find $\frac{d y}{d x}$ for each function.
$y = (2 x^{3} - x^{2} + 6 x + 1)^{3}$
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Exercises 233
For the following exercises, find $\frac{d y}{d x}$ for each function.
$y = (tan x + sin x)^{- 3}$
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Exercises 235
For the following exercises, find $\frac{d y}{d x}$ for each function.
$y = sin (cos 7 x)$
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Exercises 237
For the following exercises, find $\frac{d y}{d x}$ for each function.
$y = cot^{3} (4 x + 1)$
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Exercises 239
For the following exercises, find $\frac{d y}{d x}$ for each function.
Let $y = (f (x) + 5 x^{2})^{4}$ and suppose that and when Find
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Exercises 241
For the following exercises, find $\frac{d y}{d x}$ for each function.
[T] Find an equation of the tangent line to $y = - sin (\frac{x}{2})$ at the origin. Use a calculator to graph the function and the tangent line together.
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Exercises 243
For the following exercises, find $\frac{d y}{d x}$ for each function.
Find the $x$ -coordinates at which the tangent line to $y = (x - \frac{6}{x})^{8}$ is horizontal.
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Exercises 245
For the following exercises, use the information in the following table to find $h^{'} (a)$ at the given value for $a .$
$x$ $f (x)$ $f^{'} (x)$
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Exercises 247
For the following exercises, use the information in the following table to find $h^{'} (a)$ at the given value for $a .$
$x$ $f (x)$ $f^{'} (x)$
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Exercises 249
For the following exercises, use the information in the following table to find $h^{'} (a)$ at the given value for $a .$
$x$ $f (x)$ $f^{'} (x)$
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Exercises 251
For the following exercises, use the information in the following table to find $h^{'} (a)$ at the given value for $a .$
$x$ $f (x)$ $f^{'} (x)$
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Exercises 253
For the following exercises, use the information in the following table to find $h^{'} (a)$ at the given value for $a .$
$x$ $f (x)$ $f^{'} (x)$
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Exercises 255
For the following exercises, use the information in the following table to find $h^{'} (a)$ at the given value for $a .$
$x$ $f (x)$ $f^{'} (x)$
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Exercises 257
For the following exercises, use the information in the following table to find $h^{'} (a)$ at the given value for $a .$
$x$ $f (x)$ $f^{'} (x)$
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Exercises 259
For the following exercises, use the information in the following table to find $h^{'} (a)$ at the given value for $a .$
$x$ $f (x)$ $f^{'} (x)$
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Key terms · 1

chain rule: the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function

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The Chain Rule: practice problems

23 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 215

For the following exercises, given $y = f (u)$ and $u = g (x),$ find $\frac{d y}{d x}$ by using Leibniz’s notation for the chain rule: $\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x} .$

$y = 6 u^{3}, u = 7 x - 4$

Exercises 217

For the following exercises, given $y = f (u)$ and $u = g (x),$ find $\frac{d y}{d x}$ by using Leibniz’s notation for the chain rule: $\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x} .$

Exercises 219

For the following exercises, given $y = f (u)$ and $u = g (x),$ find $\frac{d y}{d x}$ by using Leibniz’s notation for the chain rule: $\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x} .$

Exercises 221

For each of the following exercises,

decompose each function in the form $y = f (u)$ and $u = g (x),$ and
find $\frac{d y}{d x}$ as a function of $x .$

Exercises 223

For each of the following exercises,

decompose each function in the form $y = f (u)$ and $u = g (x),$ and
find $\frac{d y}{d x}$ as a function of $x .$

Exercises 225

For each of the following exercises,

decompose each function in the form $y = f (u)$ and $u = g (x),$ and
find $\frac{d y}{d x}$ as a function of $x .$

Exercises 227

For each of the following exercises,

decompose each function in the form $y = f (u)$ and $u = g (x),$ and
find $\frac{d y}{d x}$ as a function of $x .$

Exercises 229

For the following exercises, find $\frac{d y}{d x}$ for each function.

$y = (5 - 2 x)^{- 2}$

Exercises 231

For the following exercises, find $\frac{d y}{d x}$ for each function.

$y = (2 x^{3} - x^{2} + 6 x + 1)^{3}$

Exercises 233

For the following exercises, find $\frac{d y}{d x}$ for each function.

$y = (tan x + sin x)^{- 3}$

Exercises 235

For the following exercises, find $\frac{d y}{d x}$ for each function.

$y = sin (cos 7 x)$

Exercises 237

For the following exercises, find $\frac{d y}{d x}$ for each function.

$y = cot^{3} (4 x + 1)$

Exercises 239

For the following exercises, find $\frac{d y}{d x}$ for each function.

Let $y = (f (x) + 5 x^{2})^{4}$ and suppose that and when Find

Exercises 241

For the following exercises, find $\frac{d y}{d x}$ for each function.

[T] Find an equation of the tangent line to $y = - sin (\frac{x}{2})$ at the origin. Use a calculator to graph the function and the tangent line together.

Exercises 243

For the following exercises, find $\frac{d y}{d x}$ for each function.

Find the $x$ -coordinates at which the tangent line to $y = (x - \frac{6}{x})^{8}$ is horizontal.

Exercises 245