Calculus Examples

{(x - 5)}^{3}

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

Step 1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = x^{3}

$f (x) = x^{3}$ and

g (x) = x - 5

$g (x) = x - 5$ .

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Step 1.1

To apply the Chain Rule, set

u

$u$ as

x - 5

$x - 5$ .

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

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

Step 1.2

Differentiate using the Power Rule which states that

\frac{d}{d u} [u^{n}]

$\frac{d}{d u} [u^{n}]$ is

n u^{n - 1}

$n u^{n - 1}$ where

n = 3

$n = 3$ .

3 u^{2} \frac{d}{d x} [x - 5]

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

Step 1.3

Replace all occurrences of

u

$u$ with

x - 5

$x - 5$ .

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

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

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

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of

x - 5

$x - 5$ with respect to

x

$x$ is

\frac{d}{d x} [x] + \frac{d}{d x} [- 5]

$\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

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

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

Step 2.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

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

$3 {(x - 5)}^{2} (1 + \frac{d}{d x} [- 5])$

Step 2.3

Since

- 5

$- 5$ is constant with respect to

x

$x$ , the derivative of

- 5

$- 5$ with respect to

x

$x$ is

0

$0$ .

3 {(x - 5)}^{2} (1 + 0)

$3 {(x - 5)}^{2} (1 + 0)$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add

1

$1$ and

0

$0$ .

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

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

Step 2.4.2

Multiply

3

$3$ by

1

$1$ .

3 {(x - 5)}^{2}

$3 {(x - 5)}^{2}$

3 {(x - 5)}^{2}

$3 {(x - 5)}^{2}$

3 {(x - 5)}^{2}

$3 {(x - 5)}^{2}$