Calculus Examples

$3 {(4 - 9 x)}^{4}$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 {(4 - 9 x)}^{4}$ with respect to $x$ is $3 \frac{d}{d x} [{(4 - 9 x)}^{4}]$ .

$3 \frac{d}{d x} [{(4 - 9 x)}^{4}]$

Step 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 $g (x) = 4 - 9 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $4 - 9 x$ .

$3 (\frac{d}{d u} [u^{4}] \frac{d}{d x} [4 - 9 x])$

Step 2.2

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

$3 (4 u^{3} \frac{d}{d x} [4 - 9 x])$

Step 2.3

Replace all occurrences of $u$ with $4 - 9 x$ .

$3 (4 {(4 - 9 x)}^{3} \frac{d}{d x} [4 - 9 x])$

Step 3

Differentiate.

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Step 3.1

Multiply $4$ by $3$ .

$12 ({(4 - 9 x)}^{3} \frac{d}{d x} [4 - 9 x])$

Step 3.2

By the Sum Rule, the derivative of $4 - 9 x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 9 x]$ .

$12 {(4 - 9 x)}^{3} (\frac{d}{d x} [4] + \frac{d}{d x} [- 9 x])$

Step 3.3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$12 {(4 - 9 x)}^{3} (0 + \frac{d}{d x} [- 9 x])$

Step 3.4

Add $0$ and $\frac{d}{d x} [- 9 x]$ .

$12 {(4 - 9 x)}^{3} \frac{d}{d x} [- 9 x]$

Step 3.5

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x$ with respect to $x$ is $- 9 \frac{d}{d x} [x]$ .

$12 {(4 - 9 x)}^{3} (- 9 \frac{d}{d x} [x])$

Step 3.6

Multiply $- 9$ by $12$ .

$- 108 {(4 - 9 x)}^{3} \frac{d}{d x} [x]$

Step 3.7

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

$- 108 {(4 - 9 x)}^{3} \cdot 1$

Step 3.8

Multiply $- 108$ by $1$ .

$- 108 {(4 - 9 x)}^{3}$