Calculus Examples

${(3 x - 8)}^{9}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{9}$ and $g (x) = 3 x - 8$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $3 x - 8$ .

$\frac{d}{d u} [u^{9}] \frac{d}{d x} [3 x - 8]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 9$ .

$9 u^{8} \frac{d}{d x} [3 x - 8]$

Step 1.3

Replace all occurrences of $u$ with $3 x - 8$ .

$9 {(3 x - 8)}^{8} \frac{d}{d x} [3 x - 8]$

Step 2

By the Sum Rule, the derivative of $3 x - 8$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 8]$ .

$9 {(3 x - 8)}^{8} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 8])$

Step 3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 3.1

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

$9 {(3 x - 8)}^{8} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 8])$

Step 3.2

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

$9 {(3 x - 8)}^{8} (3 \cdot 1 + \frac{d}{d x} [- 8])$

Step 3.3

Multiply $3$ by $1$ .

$9 {(3 x - 8)}^{8} (3 + \frac{d}{d x} [- 8])$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$9 {(3 x - 8)}^{8} (3 + 0)$

Step 4.2

Simplify the expression.

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Step 4.2.1

Add $3$ and $0$ .

$9 {(3 x - 8)}^{8} \cdot 3$

Step 4.2.2

Multiply $3$ by $9$ .

$27 {(3 x - 8)}^{8}$

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