Calculus Examples

$f (x) = 7 {(x^{2} - 10 x)}^{20}$

Step 1

Find the first derivative.

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Step 1.1

Since $7$ is constant with respect to $x$ , the derivative of $7 {(x^{2} - 10 x)}^{20}$ with respect to $x$ is $7 \frac{d}{d x} [{(x^{2} - 10 x)}^{20}]$ .

$7 \frac{d}{d x} [{(x^{2} - 10 x)}^{20}]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{20}$ and $g (x) = x^{2} - 10 x$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 10 x$ .

$7 (\frac{d}{d u} [u^{20}] \frac{d}{d x} [x^{2} - 10 x])$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 20$ .

$7 (20 u^{19} \frac{d}{d x} [x^{2} - 10 x])$

Step 1.2.3

Replace all occurrences of $u$ with $x^{2} - 10 x$ .

$7 (20 {(x^{2} - 10 x)}^{19} \frac{d}{d x} [x^{2} - 10 x])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $20$ by $7$ .

$140 ({(x^{2} - 10 x)}^{19} \frac{d}{d x} [x^{2} - 10 x])$

Step 1.3.2

By the Sum Rule, the derivative of $x^{2} - 10 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x]$ .

$140 {(x^{2} - 10 x)}^{19} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x])$

Step 1.3.3

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

$140 {(x^{2} - 10 x)}^{19} (2 x + \frac{d}{d x} [- 10 x])$

Step 1.3.4

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x$ with respect to $x$ is $- 10 \frac{d}{d x} [x]$ .

$140 {(x^{2} - 10 x)}^{19} (2 x - 10 \frac{d}{d x} [x])$

Step 1.3.5

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

$140 {(x^{2} - 10 x)}^{19} (2 x - 10 \cdot 1)$

Step 1.3.6

Multiply $- 10$ by $1$ .

$f' (x) = 140 {(x^{2} - 10 x)}^{19} (2 x - 10)$

Step 2

Find the second derivative.

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Step 2.1

Since $140$ is constant with respect to $x$ , the derivative of $140 {(x^{2} - 10 x)}^{19} (2 x - 10)$ with respect to $x$ is $140 \frac{d}{d x} [{(x^{2} - 10 x)}^{19} (2 x - 10)]$ .

$140 \frac{d}{d x} [{(x^{2} - 10 x)}^{19} (2 x - 10)]$

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 $f (x) = {(x^{2} - 10 x)}^{19}$ and $g (x) = 2 x - 10$ .

$140 ({(x^{2} - 10 x)}^{19} \frac{d}{d x} [2 x - 10] + (2 x - 10) \frac{d}{d x} [{(x^{2} - 10 x)}^{19}])$

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $2 x - 10$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 10]$ .

$140 ({(x^{2} - 10 x)}^{19} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 10]) + (2 x - 10) \frac{d}{d x} [{(x^{2} - 10 x)}^{19}])$

Step 2.3.2

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

$140 ({(x^{2} - 10 x)}^{19} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 10]) + (2 x - 10) \frac{d}{d x} [{(x^{2} - 10 x)}^{19}])$

Step 2.3.3

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

$140 ({(x^{2} - 10 x)}^{19} (2 \cdot 1 + \frac{d}{d x} [- 10]) + (2 x - 10) \frac{d}{d x} [{(x^{2} - 10 x)}^{19}])$

Step 2.3.4

Multiply $2$ by $1$ .

$140 ({(x^{2} - 10 x)}^{19} (2 + \frac{d}{d x} [- 10]) + (2 x - 10) \frac{d}{d x} [{(x^{2} - 10 x)}^{19}])$

Step 2.3.5

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10$ with respect to $x$ is $0$ .

$140 ({(x^{2} - 10 x)}^{19} (2 + 0) + (2 x - 10) \frac{d}{d x} [{(x^{2} - 10 x)}^{19}])$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $2$ and $0$ .

$140 ({(x^{2} - 10 x)}^{19} \cdot 2 + (2 x - 10) \frac{d}{d x} [{(x^{2} - 10 x)}^{19}])$

Step 2.3.6.2

Move $2$ to the left of ${(x^{2} - 10 x)}^{19}$ .

$140 (2 \cdot {(x^{2} - 10 x)}^{19} + (2 x - 10) \frac{d}{d x} [{(x^{2} - 10 x)}^{19}])$

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 $f (x) = x^{19}$ and $g (x) = x^{2} - 10 x$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x^{2} - 10 x$ .

$140 (2 {(x^{2} - 10 x)}^{19} + (2 x - 10) (\frac{d}{d u} [u^{19}] \frac{d}{d x} [x^{2} - 10 x]))$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 19$ .

$140 (2 {(x^{2} - 10 x)}^{19} + (2 x - 10) (19 u^{18} \frac{d}{d x} [x^{2} - 10 x]))$

Step 2.4.3

Replace all occurrences of $u$ with $x^{2} - 10 x$ .

$140 (2 {(x^{2} - 10 x)}^{19} + (2 x - 10) (19 {(x^{2} - 10 x)}^{18} \frac{d}{d x} [x^{2} - 10 x]))$

Step 2.5

Differentiate.

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Step 2.5.1

Move $19$ to the left of $2 x - 10$ .

$140 (2 {(x^{2} - 10 x)}^{19} + 19 \cdot (2 x - 10) {(x^{2} - 10 x)}^{18} \frac{d}{d x} [x^{2} - 10 x])$

Step 2.5.2

By the Sum Rule, the derivative of $x^{2} - 10 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x]$ .

$140 (2 {(x^{2} - 10 x)}^{19} + 19 (2 x - 10) {(x^{2} - 10 x)}^{18} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x]))$

Step 2.5.3

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

$140 (2 {(x^{2} - 10 x)}^{19} + 19 (2 x - 10) {(x^{2} - 10 x)}^{18} (2 x + \frac{d}{d x} [- 10 x]))$

Step 2.5.4

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x$ with respect to $x$ is $- 10 \frac{d}{d x} [x]$ .

$140 (2 {(x^{2} - 10 x)}^{19} + 19 (2 x - 10) {(x^{2} - 10 x)}^{18} (2 x - 10 \frac{d}{d x} [x]))$

Step 2.5.5

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

$140 (2 {(x^{2} - 10 x)}^{19} + 19 (2 x - 10) {(x^{2} - 10 x)}^{18} (2 x - 10 \cdot 1))$

Step 2.5.6

Multiply $- 10$ by $1$ .

$140 (2 {(x^{2} - 10 x)}^{19} + 19 (2 x - 10) {(x^{2} - 10 x)}^{18} (2 x - 10))$

Step 2.6

Raise $2 x - 10$ to the power of $1$ .

$140 (2 {(x^{2} - 10 x)}^{19} + 19 ({(2 x - 10)}^{1} (2 x - 10)) {(x^{2} - 10 x)}^{18})$

Step 2.7

Raise $2 x - 10$ to the power of $1$ .

$140 (2 {(x^{2} - 10 x)}^{19} + 19 ({(2 x - 10)}^{1} {(2 x - 10)}^{1}) {(x^{2} - 10 x)}^{18})$

Step 2.8

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

$140 (2 {(x^{2} - 10 x)}^{19} + 19 {(2 x - 10)}^{1 + 1} {(x^{2} - 10 x)}^{18})$

Step 2.9

Add $1$ and $1$ .

$140 (2 {(x^{2} - 10 x)}^{19} + 19 {(2 x - 10)}^{2} {(x^{2} - 10 x)}^{18})$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$140 (2 {(x^{2} - 10 x)}^{19}) + 140 (19 {(2 x - 10)}^{2} {(x^{2} - 10 x)}^{18})$

Step 2.10.2

Multiply $2$ by $140$ .

$280 {(x^{2} - 10 x)}^{19} + 140 (19 {(2 x - 10)}^{2} {(x^{2} - 10 x)}^{18})$

Step 2.10.3

Multiply $19$ by $140$ .

$280 {(x^{2} - 10 x)}^{19} + 2660 ({(2 x - 10)}^{2} {(x^{2} - 10 x)}^{18})$

Step 2.10.4

Factor $140 {(x^{2} - 10 x)}^{18}$ out of $280 {(x^{2} - 10 x)}^{19} + 2660 {(2 x - 10)}^{2} {(x^{2} - 10 x)}^{18}$ .

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Step 2.10.4.1

Factor $140 {(x^{2} - 10 x)}^{18}$ out of $280 {(x^{2} - 10 x)}^{19}$ .

$140 {(x^{2} - 10 x)}^{18} (2 (x^{2} - 10 x)) + 2660 {(2 x - 10)}^{2} {(x^{2} - 10 x)}^{18}$

Step 2.10.4.2

Factor $140 {(x^{2} - 10 x)}^{18}$ out of $2660 {(2 x - 10)}^{2} {(x^{2} - 10 x)}^{18}$ .

$140 {(x^{2} - 10 x)}^{18} (2 (x^{2} - 10 x)) + 140 {(x^{2} - 10 x)}^{18} (19 {(2 x - 10)}^{2})$

Step 2.10.4.3

Factor $140 {(x^{2} - 10 x)}^{18}$ out of $140 {(x^{2} - 10 x)}^{18} (2 (x^{2} - 10 x)) + 140 {(x^{2} - 10 x)}^{18} (19 {(2 x - 10)}^{2})$ .

$f'' (x) = 140 {(x^{2} - 10 x)}^{18} (2 (x^{2} - 10 x) + 19 {(2 x - 10)}^{2})$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $140 {(x^{2} - 10 x)}^{18} (2 (x^{2} - 10 x) + 19 {(2 x - 10)}^{2})$ .

$140 {(x^{2} - 10 x)}^{18} (2 (x^{2} - 10 x) + 19 {(2 x - 10)}^{2})$