Calculus Examples

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

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

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

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

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 8$ .

$8 u^{7} \frac{d}{d x} [x - 3]$

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

$8 {(x - 3)}^{7} \frac{d}{d x} [x - 3]$

Differentiate.

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

$8 {(x - 3)}^{7} (\frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

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

$8 {(x - 3)}^{7} (1 + \frac{d}{d x} [- 3])$

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$8 {(x - 3)}^{7} (1 + 0)$

Simplify the expression.

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Add $1$ and $0$ .

$8 {(x - 3)}^{7} \cdot 1$

Multiply $8$ by $1$ .

$8 {(x - 3)}^{7}$

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