Calculus Examples

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

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

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

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Step 2.1

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

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

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 $n = 2$ .

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

Multiply $2$ by $- 1$ .

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

Step 3.2

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

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

Step 3.3

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

$- 2 \sin ({(1 - x^{2})}^{2}) ((1 - x^{2}) (0 + \frac{d}{d x} [- x^{2}]))$

Step 3.4

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 3.5

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

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

Step 3.6

Multiply $- 1$ by $- 2$ .

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

Step 3.7

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

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

Step 3.8

Simplify the expression.

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Step 3.8.1

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

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

Step 3.8.2

Multiply $2$ by $2$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $x$ by $1$ .

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

Add $1$ and $2$ .

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