Calculus Examples

${(4 x - 14)}^{6}$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = 4 x - 14$ .

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To apply the Chain Rule, set $u$ as $4 x - 14$ .

$\frac{d}{d u} [u^{6}] \frac{d}{d x} [4 x - 14]$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$6 u^{5} \frac{d}{d x} [4 x - 14]$

Replace all occurrences of $u$ with $4 x - 14$ .

$6 {(4 x - 14)}^{5} \frac{d}{d x} [4 x - 14]$

Differentiate.

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By the Sum Rule, the derivative of $4 x - 14$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 14]$ .

$6 {(4 x - 14)}^{5} (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 14])$

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$6 {(4 x - 14)}^{5} (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 14])$

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

$6 {(4 x - 14)}^{5} (4 \cdot 1 + \frac{d}{d x} [- 14])$

Multiply $4$ by $1$ .

$6 {(4 x - 14)}^{5} (4 + \frac{d}{d x} [- 14])$

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14$ with respect to $x$ is $0$ .

$6 {(4 x - 14)}^{5} (4 + 0)$

Simplify the expression.

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Add $4$ and $0$ .

$6 {(4 x - 14)}^{5} \cdot 4$

Multiply $4$ by $6$ .

$24 {(4 x - 14)}^{5}$

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