Calculus Examples

$f (z) = \frac{2 \cos (z)}{z + 1}$

Step 1

Since $2$ is constant with respect to $z$ , the derivative of $\frac{2 \cos (z)}{z + 1}$ with respect to $z$ is $2 \frac{d}{d z} [\frac{\cos (z)}{z + 1}]$ .

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

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 $f (z) = \cos (z)$ and $g (z) = z + 1$ .

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

Step 3

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Combine fractions.

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Step 4.4.1

Add $1$ and $0$ .

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

Step 4.4.2

Multiply $- 1$ by $1$ .

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

Step 4.4.3

Combine $2$ and $\frac{(z + 1) (- \sin (z)) - \cos (z)}{{(z + 1)}^{2}}$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Simplify each term.

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Step 5.3.1

Rewrite using the commutative property of multiplication.

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

Step 5.3.2

Multiply $- 1$ by $2$ .

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

Step 5.3.3

Multiply $- \sin (z)$ by $1$ .

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

Step 5.3.4

Multiply $- 1$ by $2$ .

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

Step 5.3.5

Multiply $- 1$ by $2$ .

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