Calculus Examples

2 x cos (x^{2})

$2 x \cos (x^{2})$

Step 1

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 x \cos (x^{2})$ with respect to

$x$ is

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

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

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

$g (x) = \cos (x^{2})$ .

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

Step 3

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

$g (x) = x^{2}$ .

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Step 3.1

To apply the Chain Rule, set

$u$ as

$x^{2}$ .

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

Step 3.2

The derivative of

$\cos (u)$ with respect to

$u$ is

$- \sin (u)$ .

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

Step 3.3

Replace all occurrences of

$u$ with

$x^{2}$ .

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

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

$n = 2$ .

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

Step 4.2

Multiply

$2$ by

$- 1$ .

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

Step 5

Raise

$x$ to the power of

$1$ .

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

Step 6

Raise

$x$ to the power of

$1$ .

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

Step 7

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8

Add

$1$ and

$1$ .

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

Step 9

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$2 (x^{2} (- 2 \sin (x^{2})) + \cos (x^{2}) \cdot 1)$

Step 10

Multiply

$\cos (x^{2})$ by

$1$ .

$2 (x^{2} (- 2 \sin (x^{2})) + \cos (x^{2}))$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$2 (x^{2} (- 2 \sin (x^{2}))) + 2 \cos (x^{2})$

Step 11.2

Multiply

$- 2$ by

$2$ .

$- 4 x^{2} \sin (x^{2}) + 2 \cos (x^{2})$