Calculus Examples

$m (x) = {\cos (1 - x^{2})}^{\frac{3}{2}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = \cos (1 - x^{2})$ .

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Step 1.1

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

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{3}{2}}] \frac{d}{d x} [\cos (1 - x^{2})]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{3}{2} {u_{1}}^{\frac{3}{2} - 1} \frac{d}{d x} [\cos (1 - x^{2})]$

Step 1.3

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

$\frac{3}{2} {\cos (1 - x^{2})}^{\frac{3}{2} - 1} \frac{d}{d x} [\cos (1 - x^{2})]$

Step 2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} {\cos (1 - x^{2})}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [\cos (1 - x^{2})]$

Step 3

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{3}{2} {\cos (1 - x^{2})}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [\cos (1 - x^{2})]$

Step 4

Combine the numerators over the common denominator.

$\frac{3}{2} {\cos (1 - x^{2})}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} [\cos (1 - x^{2})]$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $2$ .

$\frac{3}{2} {\cos (1 - x^{2})}^{\frac{3 - 2}{2}} \frac{d}{d x} [\cos (1 - x^{2})]$

Step 5.2

Subtract $2$ from $3$ .

$\frac{3}{2} {\cos (1 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [\cos (1 - x^{2})]$

Step 6

Combine $\frac{3}{2}$ and ${\cos (1 - x^{2})}^{\frac{1}{2}}$ .

$\frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}}}{2} \frac{d}{d x} [\cos (1 - x^{2})]$

Step 7

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

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Step 7.1

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

$\frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}}}{2} (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [1 - x^{2}])$

Step 7.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$\frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}}}{2} (- \sin (u_{2}) \frac{d}{d x} [1 - x^{2}])$

Step 7.3

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

$\frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}}}{2} (- \sin (1 - x^{2}) \frac{d}{d x} [1 - x^{2}])$

Step 8

Differentiate.

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Step 8.1

Combine $\frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}}}{2}$ and $\sin (1 - x^{2})$ .

$- \frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2})}{2} \frac{d}{d x} [1 - x^{2}]$

Step 8.2

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

$- \frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2})}{2} (\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}])$

Step 8.3

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

$- \frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2})}{2} (0 + \frac{d}{d x} [- x^{2}])$

Step 8.4

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

$- \frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2})}{2} \frac{d}{d x} [- x^{2}]$

Step 8.5

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

$- \frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2})}{2} (- \frac{d}{d x} [x^{2}])$

Step 8.6

Multiply.

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Step 8.6.1

Multiply $- 1$ by $- 1$ .

$1 \frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2})}{2} \frac{d}{d x} [x^{2}]$

Step 8.6.2

Multiply $\frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2})}{2}$ by $1$ .

$\frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2})}{2} \frac{d}{d x} [x^{2}]$

Step 8.7

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

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

Step 8.8

Simplify terms.

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Step 8.8.1

Combine $2$ and $\frac{3 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2})}{2}$ .

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

Step 8.8.2

Multiply $3$ by $2$ .

$\frac{6 ({\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2}))}{2} x$

Step 8.8.3

Combine $\frac{6 ({\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2}))}{2}$ and $x$ .

$\frac{6 ({\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2})) x}{2}$

Step 8.8.4

Factor $2$ out of $6 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2}) x$ .

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

Step 9

Cancel the common factors.

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Step 9.1

Factor $2$ out of $2$ .

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

Step 9.2

Cancel the common factor.

$\frac{2 (3 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2}) x)}{2 \cdot 1}$

Step 9.3

Rewrite the expression.

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

Step 9.4

Divide $3 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2}) x$ by $1$ .

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

Step 10

Reorder the factors of $3 {\cos (1 - x^{2})}^{\frac{1}{2}} \sin (1 - x^{2}) x$ .

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