Calculus Examples

$(- \sin (- 2 x))$

Step 1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (- 2 x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (- 2 x)]$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = - 2 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $- 2 x$ .

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

Step 2.2

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

$- (\cos (u) \frac{d}{d x} [- 2 x])$

Step 2.3

Replace all occurrences of $u$ with $- 2 x$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Multiply $- 2$ by $- 1$ .

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

Step 3.3

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

$2 \cos (- 2 x) \cdot 1$

Step 3.4

Multiply $2$ by $1$ .

$2 \cos (- 2 x)$