Calculus Examples

$6 \sin (x) + 6 x \cos (x)$

By the Sum Rule, the derivative of $6 \sin (x) + 6 x \cos (x)$ with respect to $x$ is $\frac{d}{d x} [6 \sin (x)] + \frac{d}{d x} [6 x \cos (x)]$ .

$\frac{d}{d x} [6 \sin (x)] + \frac{d}{d x} [6 x \cos (x)]$

Evaluate $\frac{d}{d x} [6 \sin (x)]$ .

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Since $6$ is constant with respect to $x$ , the derivative of $6 \sin (x)$ with respect to $x$ is $6 \frac{d}{d x} [\sin (x)]$ .

$6 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [6 x \cos (x)]$

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

$6 \cos (x) + \frac{d}{d x} [6 x \cos (x)]$

Evaluate $\frac{d}{d x} [6 x \cos (x)]$ .

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Since $6$ is constant with respect to $x$ , the derivative of $6 x \cos (x)$ with respect to $x$ is $6 \frac{d}{d x} [x \cos (x)]$ .

$6 \cos (x) + 6 \frac{d}{d x} [x \cos (x)]$

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \cos (x)$ .

$6 \cos (x) + 6 (x \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x])$

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

$6 \cos (x) + 6 (x (- \sin (x)) + \cos (x) \frac{d}{d x} [x])$

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

$6 \cos (x) + 6 (x (- \sin (x)) + \cos (x) \cdot 1)$

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

$6 \cos (x) + 6 (x (- \sin (x)) + \cos (x))$

Simplify.

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Apply the distributive property.

$6 \cos (x) + 6 (x (- \sin (x))) + 6 \cos (x)$

Combine terms.

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Multiply $- 1$ by $6$ .

$6 \cos (x) - 6 (x (\sin (x))) + 6 \cos (x)$

Add $6 \cos (x)$ and $6 \cos (x)$ .

$- 6 x \sin (x) + 12 \cos (x)$