Calculus Examples

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

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 $2$ .

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