Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = 3 - 5 x$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $3 - 5 x$ .

$\frac{d}{d u} [u^{5}] \frac{d}{d x} [3 - 5 x]$

Step 1.1.2

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

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

Step 1.1.3

Replace all occurrences of $u$ with $3 - 5 x$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $3 - 5 x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- 5 x]$ .

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

Step 1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$5 {(3 - 5 x)}^{4} (0 + \frac{d}{d x} [- 5 x])$

Step 1.2.3

Add $0$ and $\frac{d}{d x} [- 5 x]$ .

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

Step 1.2.4

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

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

Step 1.2.5

Multiply $- 5$ by $5$ .

$- 25 {(3 - 5 x)}^{4} \frac{d}{d x} [x]$

Step 1.2.6

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

$- 25 {(3 - 5 x)}^{4} \cdot 1$

Step 1.2.7

Multiply $- 25$ by $1$ .

$f' (x) = - 25 {(3 - 5 x)}^{4}$

Step 2

Find the second derivative.

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Step 2.1

Since $- 25$ is constant with respect to $x$ , the derivative of $- 25 {(3 - 5 x)}^{4}$ with respect to $x$ is $- 25 \frac{d}{d x} [{(3 - 5 x)}^{4}]$ .

$- 25 \frac{d}{d x} [{(3 - 5 x)}^{4}]$

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 $f (x) = x^{4}$ and $g (x) = 3 - 5 x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $3 - 5 x$ .

$- 25 (\frac{d}{d u} [u^{4}] \frac{d}{d x} [3 - 5 x])$

Step 2.2.2

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

$- 25 (4 u^{3} \frac{d}{d x} [3 - 5 x])$

Step 2.2.3

Replace all occurrences of $u$ with $3 - 5 x$ .

$- 25 (4 {(3 - 5 x)}^{3} \frac{d}{d x} [3 - 5 x])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $4$ by $- 25$ .

$- 100 ({(3 - 5 x)}^{3} \frac{d}{d x} [3 - 5 x])$

Step 2.3.2

By the Sum Rule, the derivative of $3 - 5 x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- 5 x]$ .

$- 100 {(3 - 5 x)}^{3} (\frac{d}{d x} [3] + \frac{d}{d x} [- 5 x])$

Step 2.3.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$- 100 {(3 - 5 x)}^{3} (0 + \frac{d}{d x} [- 5 x])$

Step 2.3.4

Add $0$ and $\frac{d}{d x} [- 5 x]$ .

$- 100 {(3 - 5 x)}^{3} \frac{d}{d x} [- 5 x]$

Step 2.3.5

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

$- 100 {(3 - 5 x)}^{3} (- 5 \frac{d}{d x} [x])$

Step 2.3.6

Multiply $- 5$ by $- 100$ .

$500 {(3 - 5 x)}^{3} \frac{d}{d x} [x]$

Step 2.3.7

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

$500 {(3 - 5 x)}^{3} \cdot 1$

Step 2.3.8

Multiply $500$ by $1$ .

$f'' (x) = 500 {(3 - 5 x)}^{3}$

Step 3

Find the third derivative.

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Step 3.1

Since $500$ is constant with respect to $x$ , the derivative of $500 {(3 - 5 x)}^{3}$ with respect to $x$ is $500 \frac{d}{d x} [{(3 - 5 x)}^{3}]$ .

$500 \frac{d}{d x} [{(3 - 5 x)}^{3}]$

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 $f (x) = x^{3}$ and $g (x) = 3 - 5 x$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $3 - 5 x$ .

$500 (\frac{d}{d u} [u^{3}] \frac{d}{d x} [3 - 5 x])$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$500 (3 u^{2} \frac{d}{d x} [3 - 5 x])$

Step 3.2.3

Replace all occurrences of $u$ with $3 - 5 x$ .

$500 (3 {(3 - 5 x)}^{2} \frac{d}{d x} [3 - 5 x])$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $3$ by $500$ .

$1500 ({(3 - 5 x)}^{2} \frac{d}{d x} [3 - 5 x])$

Step 3.3.2

By the Sum Rule, the derivative of $3 - 5 x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- 5 x]$ .

$1500 {(3 - 5 x)}^{2} (\frac{d}{d x} [3] + \frac{d}{d x} [- 5 x])$

Step 3.3.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$1500 {(3 - 5 x)}^{2} (0 + \frac{d}{d x} [- 5 x])$

Step 3.3.4

Add $0$ and $\frac{d}{d x} [- 5 x]$ .

$1500 {(3 - 5 x)}^{2} \frac{d}{d x} [- 5 x]$

Step 3.3.5

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

$1500 {(3 - 5 x)}^{2} (- 5 \frac{d}{d x} [x])$

Step 3.3.6

Multiply $- 5$ by $1500$ .

$- 7500 {(3 - 5 x)}^{2} \frac{d}{d x} [x]$

Step 3.3.7

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

$- 7500 {(3 - 5 x)}^{2} \cdot 1$

Step 3.3.8

Multiply $- 7500$ by $1$ .

$f''' (x) = - 7500 {(3 - 5 x)}^{2}$