Calculus Examples

$f (x) = 12 {(3 x + 4)}^{3}$

Step 1

Since $12$ is constant with respect to $x$ , the derivative of $12 {(3 x + 4)}^{3}$ with respect to $x$ is $12 \frac{d}{d x} [{(3 x + 4)}^{3}]$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 3 x + 4$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x + 4$ .

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

Step 2.2

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

$12 (3 u^{2} \frac{d}{d x} [3 x + 4])$

Step 2.3

Replace all occurrences of $u$ with $3 x + 4$ .

$12 (3 {(3 x + 4)}^{2} \frac{d}{d x} [3 x + 4])$

Step 3

Differentiate.

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Step 3.1

Multiply $3$ by $12$ .

$36 ({(3 x + 4)}^{2} \frac{d}{d x} [3 x + 4])$

Step 3.2

By the Sum Rule, the derivative of $3 x + 4$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [4]$ .

$36 {(3 x + 4)}^{2} (\frac{d}{d x} [3 x] + \frac{d}{d x} [4])$

Step 3.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$36 {(3 x + 4)}^{2} (3 \frac{d}{d x} [x] + \frac{d}{d x} [4])$

Step 3.4

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

$36 {(3 x + 4)}^{2} (3 \cdot 1 + \frac{d}{d x} [4])$

Step 3.5

Multiply $3$ by $1$ .

$36 {(3 x + 4)}^{2} (3 + \frac{d}{d x} [4])$

Step 3.6

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

$36 {(3 x + 4)}^{2} (3 + 0)$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $3$ and $0$ .

$36 {(3 x + 4)}^{2} \cdot 3$

Step 3.7.2

Multiply $3$ by $36$ .

$108 {(3 x + 4)}^{2}$