Calculus Examples

${(2 x - 9)}^{10}$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = {(x)}^{10}$ and $g (x) = 2 x - 9$ .

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To apply the Chain Rule, set $u$ as $2 x - 9$ .

$\frac{d}{d u} [{(u)}^{10}] \frac{d}{d x} [2 x - 9]$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 10$ .

$10 u^{9} \frac{d}{d x} [2 x - 9]$

Replace all occurrences of $u$ with $2 x - 9$ .

$10 {(2 x - 9)}^{9} \frac{d}{d x} [2 x - 9]$

Differentiate.

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

$10 {(2 x - 9)}^{9} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 9])$

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

$10 {(2 x - 9)}^{9} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 9])$

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

$10 {(2 x - 9)}^{9} (2 \cdot 1 + \frac{d}{d x} [- 9])$

Multiply $2$ by $1$ .

$10 {(2 x - 9)}^{9} (2 + \frac{d}{d x} [- 9])$

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

$10 {(2 x - 9)}^{9} (2 + 0)$

Simplify the expression.

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Add $2$ and $0$ .

$10 {(2 x - 9)}^{9} \cdot 2$

Multiply $2$ by $10$ .

$20 {(2 x - 9)}^{9}$

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