Calculus Examples

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

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

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

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