Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $2 \sin (x) - \cos (2 x)$ with respect to $x$ is $\frac{d}{d x} [2 \sin (x)] + \frac{d}{d x} [- \cos (2 x)]$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Multiply $2$ by $1$ .

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

Step 3.6

Multiply $2$ by $- 1$ .

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

Step 3.7

Multiply $- 2$ by $- 1$ .

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