Precalculus Examples

$f (x) = {(6 x^{2} - 2 x + 9)}^{7}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = 6 x^{2} - 2 x + 9$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $6 x^{2} - 2 x + 9$ .

$\frac{d}{d u} [u^{7}] \frac{d}{d x} [6 x^{2} - 2 x + 9]$

Step 1.2

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

$7 u^{6} \frac{d}{d x} [6 x^{2} - 2 x + 9]$

Step 1.3

Replace all occurrences of $u$ with $6 x^{2} - 2 x + 9$ .

$7 {(6 x^{2} - 2 x + 9)}^{6} \frac{d}{d x} [6 x^{2} - 2 x + 9]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $6 x^{2} - 2 x + 9$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [9]$ .

$7 {(6 x^{2} - 2 x + 9)}^{6} (\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [9])$

Step 2.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$7 {(6 x^{2} - 2 x + 9)}^{6} (6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [9])$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$7 {(6 x^{2} - 2 x + 9)}^{6} (6 (2 x) + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [9])$

Step 2.4

Multiply $2$ by $6$ .

$7 {(6 x^{2} - 2 x + 9)}^{6} (12 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [9])$

Step 2.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$7 {(6 x^{2} - 2 x + 9)}^{6} (12 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [9])$

Step 2.6

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

$7 {(6 x^{2} - 2 x + 9)}^{6} (12 x - 2 \cdot 1 + \frac{d}{d x} [9])$

Step 2.7

Multiply $- 2$ by $1$ .

$7 {(6 x^{2} - 2 x + 9)}^{6} (12 x - 2 + \frac{d}{d x} [9])$

Step 2.8

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$7 {(6 x^{2} - 2 x + 9)}^{6} (12 x - 2 + 0)$

Step 2.9

Add $12 x - 2$ and $0$ .

$7 {(6 x^{2} - 2 x + 9)}^{6} (12 x - 2)$