Calculus Examples

$w = u^{3}$ , $u = \frac{t - 1}{t + 1}$

Step 1

The chain rule states that the derivative of $w$ with respect to $t$ is equal to the derivative of $w$ with respect to $u$ times the derivative of $u$ with respect to $t$ .

$\frac{d w}{d t} = \frac{d w}{d u} \cdot \frac{d u}{d t}$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 u^{2}$

Step 3

Find $\frac{d u}{d t}$ .

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = t - 1$ and $g (t) = t + 1$ .

$\frac{(t + 1) \frac{d}{d t} [t - 1] - (t - 1) \frac{d}{d t} [t + 1]}{{(t + 1)}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $t - 1$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [- 1]$ .

$\frac{(t + 1) (\frac{d}{d t} [t] + \frac{d}{d t} [- 1]) - (t - 1) \frac{d}{d t} [t + 1]}{{(t + 1)}^{2}}$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$\frac{(t + 1) (1 + \frac{d}{d t} [- 1]) - (t - 1) \frac{d}{d t} [t + 1]}{{(t + 1)}^{2}}$

Step 3.2.3

Since $- 1$ is constant with respect to $t$ , the derivative of $- 1$ with respect to $t$ is $0$ .

$\frac{(t + 1) (1 + 0) - (t - 1) \frac{d}{d t} [t + 1]}{{(t + 1)}^{2}}$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $1$ and $0$ .

$\frac{(t + 1) \cdot 1 - (t - 1) \frac{d}{d t} [t + 1]}{{(t + 1)}^{2}}$

Step 3.2.4.2

Multiply $t + 1$ by $1$ .

$\frac{t + 1 - (t - 1) \frac{d}{d t} [t + 1]}{{(t + 1)}^{2}}$

Step 3.2.5

By the Sum Rule, the derivative of $t + 1$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [1]$ .

$\frac{t + 1 - (t - 1) (\frac{d}{d t} [t] + \frac{d}{d t} [1])}{{(t + 1)}^{2}}$

Step 3.2.6

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$\frac{t + 1 - (t - 1) (1 + \frac{d}{d t} [1])}{{(t + 1)}^{2}}$

Step 3.2.7

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$\frac{t + 1 - (t - 1) (1 + 0)}{{(t + 1)}^{2}}$

Step 3.2.8

Simplify the expression.

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Step 3.2.8.1

Add $1$ and $0$ .

$\frac{t + 1 - (t - 1) \cdot 1}{{(t + 1)}^{2}}$

Step 3.2.8.2

Multiply $- 1$ by $1$ .

$\frac{t + 1 - (t - 1)}{{(t + 1)}^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{t + 1 - t - - 1}{{(t + 1)}^{2}}$

Step 3.3.2

Simplify the numerator.

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Step 3.3.2.1

Combine the opposite terms in $t + 1 - t - - 1$ .

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Step 3.3.2.1.1

Subtract $t$ from $t$ .

$\frac{0 + 1 - - 1}{{(t + 1)}^{2}}$

Step 3.3.2.1.2

Add $0$ and $1$ .

$\frac{1 - - 1}{{(t + 1)}^{2}}$

Step 3.3.2.2

Multiply $- 1$ by $- 1$ .

$\frac{1 + 1}{{(t + 1)}^{2}}$

Step 3.3.2.3

Add $1$ and $1$ .

$\frac{2}{{(t + 1)}^{2}}$

Step 4

Multiply $\frac{d w}{d u}$ by $\frac{d u}{d t}$ .

$\frac{d w}{d t} = (\frac{2}{{(t + 1)}^{2}}) \cdot (3 u^{2})$

Step 5

Simplify the right side $(\frac{2}{{(t + 1)}^{2}}) \cdot (3 u^{2})$ .

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Step 5.1

Rewrite using the commutative property of multiplication.

$\frac{d w}{d t} = 3 (\frac{2}{{(t + 1)}^{2}}) \cdot u^{2}$

Step 5.2

Multiply $3 \frac{2}{{(t + 1)}^{2}}$ .

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Step 5.2.1

Combine $3$ and $\frac{2}{{(t + 1)}^{2}}$ .

$\frac{d w}{d t} = \frac{3 \cdot 2}{{(t + 1)}^{2}} \cdot u^{2}$

Step 5.2.2

Multiply $3$ by $2$ .

$\frac{d w}{d t} = \frac{6}{{(t + 1)}^{2}} \cdot u^{2}$

Step 5.3

Combine $\frac{6}{{(t + 1)}^{2}}$ and $u^{2}$ .

$\frac{d w}{d t} = \frac{6 u^{2}}{{(t + 1)}^{2}}$

Step 6

Substitute in the value of $u$ into the derivative $\frac{6 u^{2}}{{(t + 1)}^{2}}$ .

$\frac{d w}{d t} = \frac{6 {(\frac{t - 1}{t + 1})}^{2}}{{(t + 1)}^{2}}$

Step 7

Simplify the result.

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Step 7.1

Apply the product rule to $\frac{t - 1}{t + 1}$ .

$\frac{d w}{d t} = \frac{6 (\frac{{(t - 1)}^{2}}{{(t + 1)}^{2}})}{{(t + 1)}^{2}}$

Step 7.2

Combine $6$ and $\frac{{(t - 1)}^{2}}{{(t + 1)}^{2}}$ .

$\frac{d w}{d t} = \frac{\frac{6 {(t - 1)}^{2}}{{(t + 1)}^{2}}}{{(t + 1)}^{2}}$

Step 7.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{d w}{d t} = \frac{6 {(t - 1)}^{2}}{{(t + 1)}^{2}} \cdot \frac{1}{{(t + 1)}^{2}}$

Step 7.4

Combine.

$\frac{d w}{d t} = \frac{6 {(t - 1)}^{2} \cdot 1}{{(t + 1)}^{2} {(t + 1)}^{2}}$

Step 7.5

Multiply ${(t + 1)}^{2}$ by ${(t + 1)}^{2}$ by adding the exponents.

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Step 7.5.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d w}{d t} = \frac{6 {(t - 1)}^{2} \cdot 1}{{(t + 1)}^{2 + 2}}$

Step 7.5.2

Add $2$ and $2$ .

$\frac{d w}{d t} = \frac{6 {(t - 1)}^{2} \cdot 1}{{(t + 1)}^{4}}$

Step 7.6

Multiply $6 {(t - 1)}^{2}$ by $1$ .

$\frac{d w}{d t} = \frac{6 {(t - 1)}^{2}}{{(t + 1)}^{4}}$