Calculus Examples

$x - \sin (2 x)$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

$1 + \frac{d}{d x} [- \sin (2 x)]$

Step 2

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

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Step 2.1

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

$1 - \frac{d}{d x} [\sin (2 x)]$

Step 2.2

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 2.2.1

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

$1 - (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x])$

Step 2.2.2

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

$1 - (\cos (u) \frac{d}{d x} [2 x])$

Step 2.2.3

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

$1 - (\cos (2 x) \frac{d}{d x} [2 x])$

Step 2.3

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

$1 - (\cos (2 x) (2 \frac{d}{d x} [x]))$

Step 2.4

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

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

Step 2.5

Multiply $2$ by $1$ .

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

Step 2.6

Move $2$ to the left of $\cos (2 x)$ .

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

Step 2.7

Multiply $2$ by $- 1$ .

$1 - 2 \cos (2 x)$