Calculus Examples

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

Find the first derivative.

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

$f' (x) = \frac{d}{d x} (\sin (x)) + \frac{d}{d x} (- \cos (x))$

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

$f' (x) = \cos (x) + \frac{d}{d x} (- \cos (x))$

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

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

$f' (x) = \cos (x) - \frac{d}{d x} \cos (x)$

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

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

Multiply $- 1$ by $- 1$ .

$f' (x) = \cos (x) + 1 \sin (x)$

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

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

Find the second derivative.

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

$f'' (x) = \frac{d}{d x} (\cos (x)) + \frac{d}{d x} (\sin (x))$

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

$f'' (x) = - \sin (x) + \frac{d}{d x} (\sin (x))$

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

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

Find the third derivative.

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

$f''' (x) = \frac{d}{d x} (- \sin (x)) + \frac{d}{d x} (\cos (x))$

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

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

$f''' (x) = - \frac{d}{d x} \sin (x) + \frac{d}{d x} (\cos (x))$

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

$f''' (x) = - \cos (x) + \frac{d}{d x} (\cos (x))$

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

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