Calculus Examples

$(2 x + 2) {(x - 5)}^{8}$

Step 1

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 $f (x) = 2 x + 2$ and $g (x) = {(x - 5)}^{8}$ .

$(2 x + 2) \frac{d}{d x} [{(x - 5)}^{8}] + {(x - 5)}^{8} \frac{d}{d x} [2 x + 2]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{8}$ and $g (x) = x - 5$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x - 5$ .

$(2 x + 2) (\frac{d}{d u} [u^{8}] \frac{d}{d x} [x - 5]) + {(x - 5)}^{8} \frac{d}{d x} [2 x + 2]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 8$ .

$(2 x + 2) (8 u^{7} \frac{d}{d x} [x - 5]) + {(x - 5)}^{8} \frac{d}{d x} [2 x + 2]$

Step 2.3

Replace all occurrences of $u$ with $x - 5$ .

$(2 x + 2) (8 {(x - 5)}^{7} \frac{d}{d x} [x - 5]) + {(x - 5)}^{8} \frac{d}{d x} [2 x + 2]$

Step 3

Differentiate.

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Step 3.1

Move $8$ to the left of $2 x + 2$ .

$8 \cdot (2 x + 2) {(x - 5)}^{7} \frac{d}{d x} [x - 5] + {(x - 5)}^{8} \frac{d}{d x} [2 x + 2]$

Step 3.2

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$8 (2 x + 2) {(x - 5)}^{7} (\frac{d}{d x} [x] + \frac{d}{d x} [- 5]) + {(x - 5)}^{8} \frac{d}{d x} [2 x + 2]$

Step 3.3

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

$8 (2 x + 2) {(x - 5)}^{7} (1 + \frac{d}{d x} [- 5]) + {(x - 5)}^{8} \frac{d}{d x} [2 x + 2]$

Step 3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$8 (2 x + 2) {(x - 5)}^{7} (1 + 0) + {(x - 5)}^{8} \frac{d}{d x} [2 x + 2]$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $1$ and $0$ .

$8 (2 x + 2) {(x - 5)}^{7} \cdot 1 + {(x - 5)}^{8} \frac{d}{d x} [2 x + 2]$

Step 3.5.2

Multiply $8$ by $1$ .

$8 (2 x + 2) {(x - 5)}^{7} + {(x - 5)}^{8} \frac{d}{d x} [2 x + 2]$

Step 3.6

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

$8 (2 x + 2) {(x - 5)}^{7} + {(x - 5)}^{8} (\frac{d}{d x} [2 x] + \frac{d}{d x} [2])$

Step 3.7

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

$8 (2 x + 2) {(x - 5)}^{7} + {(x - 5)}^{8} (2 \frac{d}{d x} [x] + \frac{d}{d x} [2])$

Step 3.8

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

$8 (2 x + 2) {(x - 5)}^{7} + {(x - 5)}^{8} (2 \cdot 1 + \frac{d}{d x} [2])$

Step 3.9

Multiply $2$ by $1$ .

$8 (2 x + 2) {(x - 5)}^{7} + {(x - 5)}^{8} (2 + \frac{d}{d x} [2])$

Step 3.10

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$8 (2 x + 2) {(x - 5)}^{7} + {(x - 5)}^{8} (2 + 0)$

Step 3.11

Simplify the expression.

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Step 3.11.1

Add $2$ and $0$ .

$8 (2 x + 2) {(x - 5)}^{7} + {(x - 5)}^{8} \cdot 2$

Step 3.11.2

Move $2$ to the left of ${(x - 5)}^{8}$ .

$8 (2 x + 2) {(x - 5)}^{7} + 2 \cdot {(x - 5)}^{8}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(8 (2 x) + 8 \cdot 2) {(x - 5)}^{7} + 2 {(x - 5)}^{8}$

Step 4.2

Multiply $2$ by $8$ .

$(16 x + 8 \cdot 2) {(x - 5)}^{7} + 2 {(x - 5)}^{8}$

Step 4.3

Multiply $8$ by $2$ .

$(16 x + 16) {(x - 5)}^{7} + 2 {(x - 5)}^{8}$

Step 4.4

Factor ${(x - 5)}^{7}$ out of $(16 x + 16) {(x - 5)}^{7} + 2 {(x - 5)}^{8}$ .

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Step 4.4.1

Factor ${(x - 5)}^{7}$ out of $(16 x + 16) {(x - 5)}^{7}$ .

${(x - 5)}^{7} (16 x + 16) + 2 {(x - 5)}^{8}$

Step 4.4.2

Factor ${(x - 5)}^{7}$ out of $2 {(x - 5)}^{8}$ .

${(x - 5)}^{7} (16 x + 16) + {(x - 5)}^{7} (2 (x - 5))$

Step 4.4.3

Factor ${(x - 5)}^{7}$ out of ${(x - 5)}^{7} (16 x + 16) + {(x - 5)}^{7} (2 (x - 5))$ .

${(x - 5)}^{7} (16 x + 16 + 2 (x - 5))$