Calculus Examples

$f (x) = \cos ({(x^{2} + 1)}^{2})$

Step 1

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 $g (x) = {(x^{2} + 1)}^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(x^{2} + 1)}^{2}$ .

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

Step 1.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with ${(x^{2} + 1)}^{2}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 1$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $x^{2} + 1$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

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

Step 2.3

Replace all occurrences of $u_{2}$ with $x^{2} + 1$ .

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

Step 3

Differentiate.

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Step 3.1

Multiply $2$ by $- 1$ .

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

Step 3.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$- 2 \sin ({(x^{2} + 1)}^{2}) ((x^{2} + 1) (2 x + 0))$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $2 x$ and $0$ .

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

Step 3.5.2

Move $2$ to the left of $x^{2} + 1$ .

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

Step 3.5.3

Multiply $2$ by $- 2$ .

$- 4 \sin ({(x^{2} + 1)}^{2}) ((x^{2} + 1) x)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$- 4 \sin ({(x^{2} + 1)}^{2}) (x^{2} x + 1 x)$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.2.1.1

Multiply $x^{2}$ by $x$ .

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Step 4.2.1.1.1

Raise $x$ to the power of $1$ .

$- 4 \sin ({(x^{2} + 1)}^{2}) (x^{2} x^{1} + 1 x)$

Step 4.2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 4 \sin ({(x^{2} + 1)}^{2}) (x^{2 + 1} + 1 x)$

Step 4.2.1.2

Add $2$ and $1$ .

$- 4 \sin ({(x^{2} + 1)}^{2}) (x^{3} + 1 x)$

Step 4.2.2

Multiply $x$ by $1$ .

$- 4 \sin ({(x^{2} + 1)}^{2}) (x^{3} + x)$