Calculus Examples

$f (x) = 4 x (\sin (x) + \cos (x))$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 x (\sin (x) + \cos (x))$ with respect to $x$ is $4 \frac{d}{d x} [x (\sin (x) + \cos (x))]$ .

$4 \frac{d}{d x} [x (\sin (x) + \cos (x))]$

Step 2

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 $g (x) = \sin (x) + \cos (x)$ .

$4 (x \frac{d}{d x} [\sin (x) + \cos (x)] + (\sin (x) + \cos (x)) \frac{d}{d x} [x])$

Step 3

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

$4 (x (\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)]) + (\sin (x) + \cos (x)) \frac{d}{d x} [x])$

Step 4

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

$4 (x (\cos (x) + \frac{d}{d x} [\cos (x)]) + (\sin (x) + \cos (x)) \frac{d}{d x} [x])$

Step 5

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

$4 (x (\cos (x) - \sin (x)) + (\sin (x) + \cos (x)) \frac{d}{d x} [x])$

Step 6

Differentiate using the Power Rule.

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Step 6.1

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

$4 (x (\cos (x) - \sin (x)) + (\sin (x) + \cos (x)) \cdot 1)$

Step 6.2

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

$4 (x (\cos (x) - \sin (x)) + \sin (x) + \cos (x))$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$4 (x \cos (x) + x (- \sin (x)) + \sin (x) + \cos (x))$

Step 7.2

Apply the distributive property.

$4 (x \cos (x)) + 4 (x (- \sin (x))) + 4 \sin (x) + 4 \cos (x)$

Step 7.3

Multiply $- 1$ by $4$ .

$4 x \cos (x) - 4 x \sin (x) + 4 \sin (x) + 4 \cos (x)$