Calculus Examples

$f (x) = \sin (\frac{z - 7}{z + 7})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d z} [f (g (z))]$ is $f' (g (z)) g' (z)$ where $f (z) = \sin (z)$ and $g (z) = \frac{z - 7}{z + 7}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{z - 7}{z + 7}$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d z} [\frac{z - 7}{z + 7}]$

Step 1.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$\cos (u) \frac{d}{d z} [\frac{z - 7}{z + 7}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{z - 7}{z + 7}$ .

$\cos (\frac{z - 7}{z + 7}) \frac{d}{d z} [\frac{z - 7}{z + 7}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d z} [\frac{f (z)}{g (z)}]$ is $\frac{g (z) \frac{d}{d z} [f (z)] - f (z) \frac{d}{d z} [g (z)]}{{g (z)}^{2}}$ where $f (z) = z - 7$ and $g (z) = z + 7$ .

$\cos (\frac{z - 7}{z + 7}) \frac{(z + 7) \frac{d}{d z} [z - 7] - (z - 7) \frac{d}{d z} [z + 7]}{{(z + 7)}^{2}}$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $z - 7$ with respect to $z$ is $\frac{d}{d z} [z] + \frac{d}{d z} [- 7]$ .

$\cos (\frac{z - 7}{z + 7}) \frac{(z + 7) (\frac{d}{d z} [z] + \frac{d}{d z} [- 7]) - (z - 7) \frac{d}{d z} [z + 7]}{{(z + 7)}^{2}}$

Step 3.2

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

$\cos (\frac{z - 7}{z + 7}) \frac{(z + 7) (1 + \frac{d}{d z} [- 7]) - (z - 7) \frac{d}{d z} [z + 7]}{{(z + 7)}^{2}}$

Step 3.3

Since $- 7$ is constant with respect to $z$ , the derivative of $- 7$ with respect to $z$ is $0$ .

$\cos (\frac{z - 7}{z + 7}) \frac{(z + 7) (1 + 0) - (z - 7) \frac{d}{d z} [z + 7]}{{(z + 7)}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

$\cos (\frac{z - 7}{z + 7}) \frac{(z + 7) \cdot 1 - (z - 7) \frac{d}{d z} [z + 7]}{{(z + 7)}^{2}}$

Step 3.4.2

Multiply $z + 7$ by $1$ .

$\cos (\frac{z - 7}{z + 7}) \frac{z + 7 - (z - 7) \frac{d}{d z} [z + 7]}{{(z + 7)}^{2}}$

Step 3.5

By the Sum Rule, the derivative of $z + 7$ with respect to $z$ is $\frac{d}{d z} [z] + \frac{d}{d z} [7]$ .

$\cos (\frac{z - 7}{z + 7}) \frac{z + 7 - (z - 7) (\frac{d}{d z} [z] + \frac{d}{d z} [7])}{{(z + 7)}^{2}}$

Step 3.6

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

$\cos (\frac{z - 7}{z + 7}) \frac{z + 7 - (z - 7) (1 + \frac{d}{d z} [7])}{{(z + 7)}^{2}}$

Step 3.7

Since $7$ is constant with respect to $z$ , the derivative of $7$ with respect to $z$ is $0$ .

$\cos (\frac{z - 7}{z + 7}) \frac{z + 7 - (z - 7) (1 + 0)}{{(z + 7)}^{2}}$

Step 3.8

Combine fractions.

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Step 3.8.1

Add $1$ and $0$ .

$\cos (\frac{z - 7}{z + 7}) \frac{z + 7 - (z - 7) \cdot 1}{{(z + 7)}^{2}}$

Step 3.8.2

Multiply $- 1$ by $1$ .

$\cos (\frac{z - 7}{z + 7}) \frac{z + 7 - (z - 7)}{{(z + 7)}^{2}}$

Step 3.8.3

Combine $\cos (\frac{z - 7}{z + 7})$ and $\frac{z + 7 - (z - 7)}{{(z + 7)}^{2}}$ .

$\frac{\cos (\frac{z - 7}{z + 7}) (z + 7 - (z - 7))}{{(z + 7)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{\cos (\frac{z - 7}{z + 7}) (z + 7 - z - - 7)}{{(z + 7)}^{2}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Combine the opposite terms in $z + 7 - z - - 7$ .

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Step 4.2.1.1

Subtract $z$ from $z$ .

$\frac{\cos (\frac{z - 7}{z + 7}) (0 + 7 - - 7)}{{(z + 7)}^{2}}$

Step 4.2.1.2

Add $0$ and $7$ .

$\frac{\cos (\frac{z - 7}{z + 7}) (7 - - 7)}{{(z + 7)}^{2}}$

Step 4.2.2

Multiply $- 1$ by $- 7$ .

$\frac{\cos (\frac{z - 7}{z + 7}) (7 + 7)}{{(z + 7)}^{2}}$

Step 4.2.3

Add $7$ and $7$ .

$\frac{\cos (\frac{z - 7}{z + 7}) \cdot 14}{{(z + 7)}^{2}}$

Step 4.2.4

Move $14$ to the left of $\cos (\frac{z - 7}{z + 7})$ .

$\frac{14 \cos (\frac{z - 7}{z + 7})}{{(z + 7)}^{2}}$