Calculus Examples

$g (x) = 6 {(5 - x)}^{7}$

Step 1

Find the first derivative.

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Step 1.1

Since $6$ is constant with respect to $x$ , the derivative of $6 {(5 - x)}^{7}$ with respect to $x$ is $6 \frac{d}{d x} [{(5 - x)}^{7}]$ .

$6 \frac{d}{d x} [{(5 - x)}^{7}]$

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 $f (x) = x^{7}$ and $g (x) = 5 - x$ .

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Step 1.2.1

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

$6 (\frac{d}{d u} [u^{7}] \frac{d}{d x} [5 - x])$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 7$ .

$6 (7 u^{6} \frac{d}{d x} [5 - x])$

Step 1.2.3

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

$6 (7 {(5 - x)}^{6} \frac{d}{d x} [5 - x])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $7$ by $6$ .

$42 ({(5 - x)}^{6} \frac{d}{d x} [5 - x])$

Step 1.3.2

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

$42 {(5 - x)}^{6} (\frac{d}{d x} [5] + \frac{d}{d x} [- x])$

Step 1.3.3

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

$42 {(5 - x)}^{6} (0 + \frac{d}{d x} [- x])$

Step 1.3.4

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

$42 {(5 - x)}^{6} \frac{d}{d x} [- x]$

Step 1.3.5

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

$42 {(5 - x)}^{6} (- \frac{d}{d x} [x])$

Step 1.3.6

Multiply $- 1$ by $42$ .

$- 42 {(5 - x)}^{6} \frac{d}{d x} [x]$

Step 1.3.7

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

$- 42 {(5 - x)}^{6} \cdot 1$

Step 1.3.8

Multiply $- 42$ by $1$ .

$f' (x) = - 42 {(5 - x)}^{6}$

Step 2

Find the second derivative.

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Step 2.1

Since $- 42$ is constant with respect to $x$ , the derivative of $- 42 {(5 - x)}^{6}$ with respect to $x$ is $- 42 \frac{d}{d x} [{(5 - x)}^{6}]$ .

$- 42 \frac{d}{d x} [{(5 - x)}^{6}]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = 5 - x$ .

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Step 2.2.1

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

$- 42 (\frac{d}{d u} [u^{6}] \frac{d}{d x} [5 - x])$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$- 42 (6 u^{5} \frac{d}{d x} [5 - x])$

Step 2.2.3

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

$- 42 (6 {(5 - x)}^{5} \frac{d}{d x} [5 - x])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $6$ by $- 42$ .

$- 252 ({(5 - x)}^{5} \frac{d}{d x} [5 - x])$

Step 2.3.2

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

$- 252 {(5 - x)}^{5} (\frac{d}{d x} [5] + \frac{d}{d x} [- x])$

Step 2.3.3

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

$- 252 {(5 - x)}^{5} (0 + \frac{d}{d x} [- x])$

Step 2.3.4

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

$- 252 {(5 - x)}^{5} \frac{d}{d x} [- x]$

Step 2.3.5

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

$- 252 {(5 - x)}^{5} (- \frac{d}{d x} [x])$

Step 2.3.6

Multiply $- 1$ by $- 252$ .

$252 {(5 - x)}^{5} \frac{d}{d x} [x]$

Step 2.3.7

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

$252 {(5 - x)}^{5} \cdot 1$

Step 2.3.8

Multiply $252$ by $1$ .

$f'' (x) = 252 {(5 - x)}^{5}$

Step 3

Find the third derivative.

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Step 3.1

Since $252$ is constant with respect to $x$ , the derivative of $252 {(5 - x)}^{5}$ with respect to $x$ is $252 \frac{d}{d x} [{(5 - x)}^{5}]$ .

$252 \frac{d}{d x} [{(5 - x)}^{5}]$

Step 3.2

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

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Step 3.2.1

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

$252 (\frac{d}{d u} [u^{5}] \frac{d}{d x} [5 - x])$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$252 (5 u^{4} \frac{d}{d x} [5 - x])$

Step 3.2.3

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

$252 (5 {(5 - x)}^{4} \frac{d}{d x} [5 - x])$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $5$ by $252$ .

$1260 ({(5 - x)}^{4} \frac{d}{d x} [5 - x])$

Step 3.3.2

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

$1260 {(5 - x)}^{4} (\frac{d}{d x} [5] + \frac{d}{d x} [- x])$

Step 3.3.3

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

$1260 {(5 - x)}^{4} (0 + \frac{d}{d x} [- x])$

Step 3.3.4

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

$1260 {(5 - x)}^{4} \frac{d}{d x} [- x]$

Step 3.3.5

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

$1260 {(5 - x)}^{4} (- \frac{d}{d x} [x])$

Step 3.3.6

Multiply $- 1$ by $1260$ .

$- 1260 {(5 - x)}^{4} \frac{d}{d x} [x]$

Step 3.3.7

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

$- 1260 {(5 - x)}^{4} \cdot 1$

Step 3.3.8

Multiply $- 1260$ by $1$ .

$f''' (x) = - 1260 {(5 - x)}^{4}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $- 1260$ is constant with respect to $x$ , the derivative of $- 1260 {(5 - x)}^{4}$ with respect to $x$ is $- 1260 \frac{d}{d x} [{(5 - x)}^{4}]$ .

$- 1260 \frac{d}{d x} [{(5 - x)}^{4}]$

Step 4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 5 - x$ .

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Step 4.2.1

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

$- 1260 (\frac{d}{d u} [u^{4}] \frac{d}{d x} [5 - x])$

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$- 1260 (4 u^{3} \frac{d}{d x} [5 - x])$

Step 4.2.3

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

$- 1260 (4 {(5 - x)}^{3} \frac{d}{d x} [5 - x])$

Step 4.3

Differentiate.

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Step 4.3.1

Multiply $4$ by $- 1260$ .

$- 5040 ({(5 - x)}^{3} \frac{d}{d x} [5 - x])$

Step 4.3.2

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

$- 5040 {(5 - x)}^{3} (\frac{d}{d x} [5] + \frac{d}{d x} [- x])$

Step 4.3.3

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

$- 5040 {(5 - x)}^{3} (0 + \frac{d}{d x} [- x])$

Step 4.3.4

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

$- 5040 {(5 - x)}^{3} \frac{d}{d x} [- x]$

Step 4.3.5

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

$- 5040 {(5 - x)}^{3} (- \frac{d}{d x} [x])$

Step 4.3.6

Multiply $- 1$ by $- 5040$ .

$5040 {(5 - x)}^{3} \frac{d}{d x} [x]$

Step 4.3.7

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

$5040 {(5 - x)}^{3} \cdot 1$

Step 4.3.8

Multiply $5040$ by $1$ .

$f^{4} (x) = 5040 {(5 - x)}^{3}$

Step 5

The fourth derivative of $g (x)$ with respect to $x$ is $5040 {(5 - x)}^{3}$ .

$5040 {(5 - x)}^{3}$