Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Suppose $u (x) = \sqrt{3 x^{2} + 1}$ and $f (u) = \frac{u^{2} + 3 u^{5}}{1 - u}$ . Use the chain rule to find role="math" localid="1648356625815" $\frac{d}{d x} (f (u (x)))$ without first finding the formula for $f (u (x))$ .

The derivative of the function is: $\frac{3 x (45 x^{2} \sqrt{3 x^{2} + 1} + 14 \sqrt{3 x^{2} + 1} - 36 x^{4} - 72 x^{2} - 10)}{{(1 - \sqrt{3 x^{2} + 1})}^{2}}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information:

The functions are:

$u (x) = \sqrt{3 x^{2} + 1}$

$f = \frac{u^{2} + 3 u^{5}}{1 - u}$

The composite function is: $f (u (x))$

Step 2. Find derivative of u(x) using chain rule.

Since, $u (x) = \sqrt{3 x^{2} + 1} u (x) = {(3 x^{2} + 1)}^{\frac{1}{2}}$

Let $v = 3 x^{2} + 1$

Then $u = {(v)}^{\frac{1}{2}}$

Derivative of $u (v)$ with respect to $v$ :

$\frac{d u}{d v} = \frac{1}{2} {(v)}^{\frac{1}{2} - 1} = \frac{1}{2} {(v)}^{- \frac{1}{2}} = \frac{1}{2 \sqrt{v}} = \frac{1}{2 \sqrt{3 x^{2} + 1}}$

Derivative of $v (x)$ with respect to $x$ :

$\frac{d v}{d x} = \frac{d}{d x} (3 x^{2} + 1) = 3.2 x + 0 = 6 x$

$\frac{d u}{d x} = \frac{d u}{d v} \times \frac{d v}{d x} = \frac{1}{2 \sqrt{3 x^{2} + 1}} \times 6 x = \frac{3 x}{\sqrt{3 x^{2} + 1}}$

Step 3. Find the derivative of f(u):

$f (u) = \frac{u^{2} + 3 u^{5}}{1 - u}$

Differentiate $\frac{d f}{d u} = \frac{(1 - u) \frac{d}{d u} (u^{2} + 3 u^{5}) - (u^{2} + 3 u^{5}) \frac{d}{d u} (1 - u)}{{(1 - u)}^{2}} = \frac{(1 - u) (2 u + 15 u^{4}) - (u^{2} + 3 u^{5}) (- 1)}{{(1 - u)}^{2}} = \frac{2 u + 15 u^{4} - 2 u^{2} - 15 u^{5} + u^{2} + 3 u^{5}}{{(1 - u)}^{2}} = \frac{2 u + 15 u^{4} - u^{2} - 12 u^{5}}{{(1 - u)}^{2}}$

Step 4. Find derivative of f(u(x)):

By using the chain rule, the derivative of the function is:

$\frac{d}{d x} (f (u (x))) = \frac{d f}{d u} \times \frac{d u}{d x}$

Substitute the values from step 3 and 4.

$\frac{d}{d x} (f (u (x))) = \frac{2 u + 15 u^{4} - u^{2} - 12 u^{5}}{{(1 - u)}^{2}} \times \frac{3 x}{\sqrt{3 x^{2} + 1}}$

Substitute value of $u (x) = \sqrt{3 x^{2} + 1}$ and simplify:

$\frac{d}{d x} (f (u (x))) = \frac{2 \sqrt{3 x^{2} + 1} + 15 {(\sqrt{3 x^{2} + 1})}^{4} - {(\sqrt{3 x^{2} + 1})}^{2} - 12 {(\sqrt{3 x^{2} + 1})}^{5}}{{(1 - \sqrt{3 x^{2} + 1})}^{2}} \times \frac{3 x}{\sqrt{3 x^{2} + 1}} = \frac{\sqrt{3 x^{2} + 1} (2 + 15 {(\sqrt{3 x^{2} + 1})}^{3} - \sqrt{3 x^{2} + 1} - 12 {(\sqrt{3 x^{2} + 1})}^{4})}{{(1 - \sqrt{3 x^{2} + 1})}^{2}} \times \frac{3 x}{\sqrt{3 x^{2} + 1}} = \frac{3 x (2 + 15 {(\sqrt{3 x^{2} + 1})}^{3} - \sqrt{3 x^{2} + 1} - 12 {(\sqrt{3 x^{2} + 1})}^{4})}{{(1 - \sqrt{3 x^{2} + 1})}^{2}} = \frac{3 x (2 + 15 (3 x^{2} + 1) \sqrt{3 x^{2} + 1} - \sqrt{3 x^{2} + 1} - 12 {(3 x^{2} + 1)}^{2})}{{(1 - \sqrt{3 x^{2} + 1})}^{2}} = \frac{3 x (2 + 45 x^{2} \sqrt{3 x^{2} + 1} + 15 \sqrt{3 x^{2} + 1} - \sqrt{3 x^{2} + 1} - 12 (3 x^{4} + 1 + 6 x^{2}))}{{(1 - \sqrt{3 x^{2} + 1})}^{2}} = \frac{3 x (2 + 45 x^{2} \sqrt{3 x^{2} + 1} + 14 \sqrt{3 x^{2} + 1} - 36 x^{4} - 12 - 72 x^{2})}{{(1 - \sqrt{3 x^{2} + 1})}^{2}} = \frac{3 x (45 x^{2} \sqrt{3 x^{2} + 1} + 14 \sqrt{3 x^{2} + 1} - 36 x^{4} - 72 x^{2} - 10)}{{(1 - \sqrt{3 x^{2} + 1})}^{2}}$

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Calculus

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Short Answer

Suppose $u (x) = \sqrt{3 x^{2} + 1}$ and $f (u) = \frac{u^{2} + 3 u^{5}}{1 - u}$ . Use the chain rule to find role="math" localid="1648356625815" $\frac{d}{d x} (f (u (x)))$ without first finding the formula for $f (u (x))$ .

Step by Step Solution

Step 1. Given information:

Step 2. Find derivative of u(x) using chain rule.

Step 3. Find the derivative of f(u):

Step 4. Find derivative of f(u(x)):

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Suppose u(x)=3x2+1 and f(u)=u2+3u51-u. Use the chain rule to find role="math" localid="1648356625815" d dx (f(u(x))) without first finding the formula for f(u(x)).

Step by Step Solution

Step 1. Given information:

Step 2. Find derivative of u(x) using chain rule.

Step 3. Find the derivative of f(u):

Step 4. Find derivative of f(u(x)):

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Suppose $u (x) = \sqrt{3 x^{2} + 1}$ and $f (u) = \frac{u^{2} + 3 u^{5}}{1 - u}$ . Use the chain rule to find role="math" localid="1648356625815" $\frac{d}{d x} (f (u (x)))$ without first finding the formula for $f (u (x))$ .