Calculus Examples

$f (x) = \sin ({(0.5 - x)}^{1 + a})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = {(0.5 - x)}^{1 + a}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(0.5 - x)}^{1 + a}$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [{(0.5 - x)}^{1 + a}]$

Step 1.2

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

$\cos (u_{1}) \frac{d}{d x} [{(0.5 - x)}^{1 + a}]$

Step 1.3

Replace all occurrences of $u_{1}$ with ${(0.5 - x)}^{1 + a}$ .

$\cos ({(0.5 - x)}^{1 + a}) \frac{d}{d x} [{(0.5 - x)}^{1 + a}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{1 + a}$ and $g (x) = 0.5 - x$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $0.5 - x$ .

$\cos ({(0.5 - x)}^{1 + a}) (\frac{d}{d u_{2}} [{u_{2}}^{1 + a}] \frac{d}{d x} [0.5 - x])$

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 $n = 1 + a$ .

$\cos ({(0.5 - x)}^{1 + a}) ((1 + a) {u_{2}}^{1 + a - 1} \frac{d}{d x} [0.5 - x])$

Step 2.3

Replace all occurrences of $u_{2}$ with $0.5 - x$ .

$\cos ({(0.5 - x)}^{1 + a}) ((1 + a) {(0.5 - x)}^{1 + a - 1} \frac{d}{d x} [0.5 - x])$

Step 3

Differentiate.

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Step 3.1

Subtract $1$ from $1$ .

$\cos ({(0.5 - x)}^{1 + a}) ((1 + a) {(0.5 - x)}^{a + 0} \frac{d}{d x} [0.5 - x])$

Step 3.2

Add $a$ and $0$ .

$\cos ({(0.5 - x)}^{1 + a}) ((1 + a) {(0.5 - x)}^{a} \frac{d}{d x} [0.5 - x])$

Step 3.3

By the Sum Rule, the derivative of $0.5 - x$ with respect to $x$ is $\frac{d}{d x} [0.5] + \frac{d}{d x} [- x]$ .

$\cos ({(0.5 - x)}^{1 + a}) ((1 + a) {(0.5 - x)}^{a} (\frac{d}{d x} [0.5] + \frac{d}{d x} [- x]))$

Step 3.4

Since $0.5$ is constant with respect to $x$ , the derivative of $0.5$ with respect to $x$ is $0$ .

$\cos ({(0.5 - x)}^{1 + a}) ((1 + a) {(0.5 - x)}^{a} (0 + \frac{d}{d x} [- x]))$

Step 3.5

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

$\cos ({(0.5 - x)}^{1 + a}) ((1 + a) {(0.5 - x)}^{a} \frac{d}{d x} [- x])$

Step 3.6

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

$\cos ({(0.5 - x)}^{1 + a}) ((1 + a) {(0.5 - x)}^{a} (- \frac{d}{d x} [x]))$

Step 3.7

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

$\cos ({(0.5 - x)}^{1 + a}) ((1 + a) {(0.5 - x)}^{a} (- 1 \cdot 1))$

Step 3.8

Simplify the expression.

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Step 3.8.1

Multiply $- 1$ by $1$ .

$\cos ({(0.5 - x)}^{1 + a}) ((1 + a) {(0.5 - x)}^{a} \cdot - 1)$

Step 3.8.2

Move $- 1$ to the left of $(1 + a) {(0.5 - x)}^{a}$ .

$\cos ({(0.5 - x)}^{1 + a}) (- 1 \cdot ((1 + a) {(0.5 - x)}^{a}))$

Step 3.8.3

Rewrite $- 1 (1 + a)$ as $- (1 + a)$ .

$\cos ({(0.5 - x)}^{1 + a}) (- (1 + a) {(0.5 - x)}^{a})$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\cos ({(0.5 - x)}^{1 + a}) ((- 1 \cdot 1 - a) {(0.5 - x)}^{a})$

Step 4.2

Multiply $- 1$ by $1$ .

$\cos ({(0.5 - x)}^{1 + a}) ((- 1 - a) {(0.5 - x)}^{a})$

Step 4.3

Reorder the factors of $\cos ({(0.5 - x)}^{1 + a}) ((- 1 - a) {(0.5 - x)}^{a})$ .

${(0.5 - x)}^{a} \cos ({(0.5 - x)}^{1 + a}) (- 1 - a)$