Calculus Examples

$2 x \sin (x)$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 x \sin (x)$ with respect to $x$ is $2 \frac{d}{d x} [x \sin (x)]$ .

$2 \frac{d}{d x} [x \sin (x)]$

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

$2 (x \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x])$

Step 3

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

$2 (x \cos (x) + \sin (x) \frac{d}{d x} [x])$

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$2 (x \cos (x) + \sin (x) \cdot 1)$

Step 4.2

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

$2 (x \cos (x) + \sin (x))$

Step 5

Apply the distributive property.

$2 x \cos (x) + 2 \sin (x)$