Calculus Examples

$\sin (x) + \cos (x)$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\sin (x) + \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)]$ .

$\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)]$

Step 1.2

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

$\cos (x) + \frac{d}{d x} [\cos (x)]$

Step 1.3

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

$f' (x) = \cos (x) - \sin (x)$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\cos (x) - \sin (x)$ with respect to $x$ is $\frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- \sin (x)]$ .

$\frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- \sin (x)]$

Step 2.2

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

$- \sin (x) + \frac{d}{d x} [- \sin (x)]$

Step 2.3

Evaluate $\frac{d}{d x} [- \sin (x)]$ .

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Step 2.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (x)]$ .

$- \sin (x) - \frac{d}{d x} [\sin (x)]$

Step 2.3.2

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

$f'' (x) = - \sin (x) - \cos (x)$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $- \sin (x) - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [- \sin (x)] + \frac{d}{d x} [- \cos (x)]$ .

$\frac{d}{d x} [- \sin (x)] + \frac{d}{d x} [- \cos (x)]$

Step 3.2

Evaluate $\frac{d}{d x} [- \sin (x)]$ .

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Step 3.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (x)]$ .

$- \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \cos (x)]$

Step 3.2.2

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

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

Step 3.3

Evaluate $\frac{d}{d x} [- \cos (x)]$ .

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Step 3.3.1

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

$- \cos (x) - \frac{d}{d x} [\cos (x)]$

Step 3.3.2

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

$- \cos (x) - - \sin (x)$

Step 3.3.3

Multiply $- 1$ by $- 1$ .

$- \cos (x) + 1 \sin (x)$

Step 3.3.4

Multiply $\sin (x)$ by $1$ .

$f''' (x) = - \cos (x) + \sin (x)$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $- \cos (x) + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [- \cos (x)] + \frac{d}{d x} [\sin (x)]$ .

$\frac{d}{d x} [- \cos (x)] + \frac{d}{d x} [\sin (x)]$

Step 4.2

Evaluate $\frac{d}{d x} [- \cos (x)]$ .

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Step 4.2.1

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

$- \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [\sin (x)]$

Step 4.2.2

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

$- - \sin (x) + \frac{d}{d x} [\sin (x)]$

Step 4.2.3

Multiply $- 1$ by $- 1$ .

$1 \sin (x) + \frac{d}{d x} [\sin (x)]$

Step 4.2.4

Multiply $\sin (x)$ by $1$ .

$\sin (x) + \frac{d}{d x} [\sin (x)]$

Step 4.3

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

$f^{4} (x) = \sin (x) + \cos (x)$