Calculus Examples

$g (x) = 8 x {(2 x - 5)}^{9}$

Step 1

Find the first derivative.

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Step 1.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x {(2 x - 5)}^{9}$ with respect to $x$ is $8 \frac{d}{d x} [x {(2 x - 5)}^{9}]$ .

$8 \frac{d}{d x} [x {(2 x - 5)}^{9}]$

Step 1.2

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 $f (x) = x$ and $g (x) = {(2 x - 5)}^{9}$ .

$8 (x \frac{d}{d x} [{(2 x - 5)}^{9}] + {(2 x - 5)}^{9} \frac{d}{d x} [x])$

Step 1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{9}$ and $g (x) = 2 x - 5$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $2 x - 5$ .

$8 (x (\frac{d}{d u} [u^{9}] \frac{d}{d x} [2 x - 5]) + {(2 x - 5)}^{9} \frac{d}{d x} [x])$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 9$ .

$8 (x (9 u^{8} \frac{d}{d x} [2 x - 5]) + {(2 x - 5)}^{9} \frac{d}{d x} [x])$

Step 1.3.3

Replace all occurrences of $u$ with $2 x - 5$ .

$8 (x (9 {(2 x - 5)}^{8} \frac{d}{d x} [2 x - 5]) + {(2 x - 5)}^{9} \frac{d}{d x} [x])$

Step 1.4

Differentiate.

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Step 1.4.1

By the Sum Rule, the derivative of $2 x - 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]$ .

$8 (x (9 {(2 x - 5)}^{8} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5])) + {(2 x - 5)}^{9} \frac{d}{d x} [x])$

Step 1.4.2

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

$8 (x (9 {(2 x - 5)}^{8} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 5])) + {(2 x - 5)}^{9} \frac{d}{d x} [x])$

Step 1.4.3

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

$8 (x (9 {(2 x - 5)}^{8} (2 \cdot 1 + \frac{d}{d x} [- 5])) + {(2 x - 5)}^{9} \frac{d}{d x} [x])$

Step 1.4.4

Multiply $2$ by $1$ .

$8 (x (9 {(2 x - 5)}^{8} (2 + \frac{d}{d x} [- 5])) + {(2 x - 5)}^{9} \frac{d}{d x} [x])$

Step 1.4.5

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

$8 (x (9 {(2 x - 5)}^{8} (2 + 0)) + {(2 x - 5)}^{9} \frac{d}{d x} [x])$

Step 1.4.6

Simplify the expression.

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Step 1.4.6.1

Add $2$ and $0$ .

$8 (x (9 {(2 x - 5)}^{8} \cdot 2) + {(2 x - 5)}^{9} \frac{d}{d x} [x])$

Step 1.4.6.2

Multiply $2$ by $9$ .

$8 (x (18 {(2 x - 5)}^{8}) + {(2 x - 5)}^{9} \frac{d}{d x} [x])$

Step 1.4.7

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

$8 (x (18 {(2 x - 5)}^{8}) + {(2 x - 5)}^{9} \cdot 1)$

Step 1.4.8

Multiply ${(2 x - 5)}^{9}$ by $1$ .

$8 (x (18 {(2 x - 5)}^{8}) + {(2 x - 5)}^{9})$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

$8 (x (18 {(2 x - 5)}^{8})) + 8 {(2 x - 5)}^{9}$

Step 1.5.2

Multiply $18$ by $8$ .

$144 (x ({(2 x - 5)}^{8})) + 8 {(2 x - 5)}^{9}$

Step 1.5.3

Factor $8 {(2 x - 5)}^{8}$ out of $144 x {(2 x - 5)}^{8} + 8 {(2 x - 5)}^{9}$ .

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Step 1.5.3.1

Factor $8 {(2 x - 5)}^{8}$ out of $144 x {(2 x - 5)}^{8}$ .

$8 {(2 x - 5)}^{8} (18 x) + 8 {(2 x - 5)}^{9}$

Step 1.5.3.2

Factor $8 {(2 x - 5)}^{8}$ out of $8 {(2 x - 5)}^{9}$ .

$8 {(2 x - 5)}^{8} (18 x) + 8 {(2 x - 5)}^{8} (2 x - 5)$

Step 1.5.3.3

Factor $8 {(2 x - 5)}^{8}$ out of $8 {(2 x - 5)}^{8} (18 x) + 8 {(2 x - 5)}^{8} (2 x - 5)$ .

$8 {(2 x - 5)}^{8} (18 x + 2 x - 5)$

Step 1.5.4

Add $18 x$ and $2 x$ .

$f' (x) = 8 {(2 x - 5)}^{8} (20 x - 5)$

Step 2

Find the second derivative.

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Step 2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 {(2 x - 5)}^{8} (20 x - 5)$ with respect to $x$ is $8 \frac{d}{d x} [{(2 x - 5)}^{8} (20 x - 5)]$ .

$8 \frac{d}{d x} [{(2 x - 5)}^{8} (20 x - 5)]$

Step 2.2

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 $f (x) = {(2 x - 5)}^{8}$ and $g (x) = 20 x - 5$ .

$8 ({(2 x - 5)}^{8} \frac{d}{d x} [20 x - 5] + (20 x - 5) \frac{d}{d x} [{(2 x - 5)}^{8}])$

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $20 x - 5$ with respect to $x$ is $\frac{d}{d x} [20 x] + \frac{d}{d x} [- 5]$ .

$8 ({(2 x - 5)}^{8} (\frac{d}{d x} [20 x] + \frac{d}{d x} [- 5]) + (20 x - 5) \frac{d}{d x} [{(2 x - 5)}^{8}])$

Step 2.3.2

Since $20$ is constant with respect to $x$ , the derivative of $20 x$ with respect to $x$ is $20 \frac{d}{d x} [x]$ .

$8 ({(2 x - 5)}^{8} (20 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]) + (20 x - 5) \frac{d}{d x} [{(2 x - 5)}^{8}])$

Step 2.3.3

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

$8 ({(2 x - 5)}^{8} (20 \cdot 1 + \frac{d}{d x} [- 5]) + (20 x - 5) \frac{d}{d x} [{(2 x - 5)}^{8}])$

Step 2.3.4

Multiply $20$ by $1$ .

$8 ({(2 x - 5)}^{8} (20 + \frac{d}{d x} [- 5]) + (20 x - 5) \frac{d}{d x} [{(2 x - 5)}^{8}])$

Step 2.3.5

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

$8 ({(2 x - 5)}^{8} (20 + 0) + (20 x - 5) \frac{d}{d x} [{(2 x - 5)}^{8}])$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $20$ and $0$ .

$8 ({(2 x - 5)}^{8} \cdot 20 + (20 x - 5) \frac{d}{d x} [{(2 x - 5)}^{8}])$

Step 2.3.6.2

Move $20$ to the left of ${(2 x - 5)}^{8}$ .

$8 (20 \cdot {(2 x - 5)}^{8} + (20 x - 5) \frac{d}{d x} [{(2 x - 5)}^{8}])$

Step 2.4

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 $g (x) = 2 x - 5$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $2 x - 5$ .

$8 (20 {(2 x - 5)}^{8} + (20 x - 5) (\frac{d}{d u} [u^{8}] \frac{d}{d x} [2 x - 5]))$

Step 2.4.2

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

$8 (20 {(2 x - 5)}^{8} + (20 x - 5) (8 u^{7} \frac{d}{d x} [2 x - 5]))$

Step 2.4.3

Replace all occurrences of $u$ with $2 x - 5$ .

$8 (20 {(2 x - 5)}^{8} + (20 x - 5) (8 {(2 x - 5)}^{7} \frac{d}{d x} [2 x - 5]))$

Step 2.5

Differentiate.

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Step 2.5.1

Move $8$ to the left of $20 x - 5$ .

$8 (20 {(2 x - 5)}^{8} + 8 \cdot (20 x - 5) {(2 x - 5)}^{7} \frac{d}{d x} [2 x - 5])$

Step 2.5.2

By the Sum Rule, the derivative of $2 x - 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]$ .

$8 (20 {(2 x - 5)}^{8} + 8 (20 x - 5) {(2 x - 5)}^{7} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]))$

Step 2.5.3

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

$8 (20 {(2 x - 5)}^{8} + 8 (20 x - 5) {(2 x - 5)}^{7} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]))$

Step 2.5.4

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

$8 (20 {(2 x - 5)}^{8} + 8 (20 x - 5) {(2 x - 5)}^{7} (2 \cdot 1 + \frac{d}{d x} [- 5]))$

Step 2.5.5

Multiply $2$ by $1$ .

$8 (20 {(2 x - 5)}^{8} + 8 (20 x - 5) {(2 x - 5)}^{7} (2 + \frac{d}{d x} [- 5]))$

Step 2.5.6

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

$8 (20 {(2 x - 5)}^{8} + 8 (20 x - 5) {(2 x - 5)}^{7} (2 + 0))$

Step 2.5.7

Simplify the expression.

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Step 2.5.7.1

Add $2$ and $0$ .

$8 (20 {(2 x - 5)}^{8} + 8 (20 x - 5) {(2 x - 5)}^{7} \cdot 2)$

Step 2.5.7.2

Multiply $2$ by $8$ .

$8 (20 {(2 x - 5)}^{8} + 16 (20 x - 5) {(2 x - 5)}^{7})$

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

$8 (20 {(2 x - 5)}^{8} + (16 (20 x) + 16 \cdot - 5) {(2 x - 5)}^{7})$

Step 2.6.2

Apply the distributive property.

$8 (20 {(2 x - 5)}^{8}) + 8 ((16 (20 x) + 16 \cdot - 5) {(2 x - 5)}^{7})$

Step 2.6.3

Multiply $20$ by $8$ .

$160 {(2 x - 5)}^{8} + 8 ((16 (20 x) + 16 \cdot - 5) {(2 x - 5)}^{7})$

Step 2.6.4

Multiply $20$ by $16$ .

$160 {(2 x - 5)}^{8} + 8 ((320 x + 16 \cdot - 5) {(2 x - 5)}^{7})$

Step 2.6.5

Multiply $16$ by $- 5$ .

$160 {(2 x - 5)}^{8} + 8 ((320 x - 80) {(2 x - 5)}^{7})$

Step 2.6.6

Factor $8 {(2 x - 5)}^{7}$ out of $160 {(2 x - 5)}^{8} + 8 (320 x - 80) {(2 x - 5)}^{7}$ .

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Step 2.6.6.1

Factor $8 {(2 x - 5)}^{7}$ out of $160 {(2 x - 5)}^{8}$ .

$8 {(2 x - 5)}^{7} (20 (2 x - 5)) + 8 (320 x - 80) {(2 x - 5)}^{7}$

Step 2.6.6.2

Factor $8 {(2 x - 5)}^{7}$ out of $8 (320 x - 80) {(2 x - 5)}^{7}$ .

$8 {(2 x - 5)}^{7} (20 (2 x - 5)) + 8 {(2 x - 5)}^{7} (320 x - 80)$

Step 2.6.6.3

Factor $8 {(2 x - 5)}^{7}$ out of $8 {(2 x - 5)}^{7} (20 (2 x - 5)) + 8 {(2 x - 5)}^{7} (320 x - 80)$ .

$f'' (x) = 8 {(2 x - 5)}^{7} (20 (2 x - 5) + 320 x - 80)$

Step 3

The second derivative of $g (x)$ with respect to $x$ is $8 {(2 x - 5)}^{7} (20 (2 x - 5) + 320 x - 80)$ .

$8 {(2 x - 5)}^{7} (20 (2 x - 5) + 320 x - 80)$