Calculus Examples

$f (x) = {(14 x - x^{2})}^{3}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 14 x - x^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $14 x - x^{2}$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [14 x - x^{2}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 u^{2} \frac{d}{d x} [14 x - x^{2}]$

Step 1.3

Replace all occurrences of $u$ with $14 x - x^{2}$ .

$3 {(14 x - x^{2})}^{2} \frac{d}{d x} [14 x - x^{2}]$

Step 2

By the Sum Rule, the derivative of $14 x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [14 x] + \frac{d}{d x} [- x^{2}]$ .

$3 {(14 x - x^{2})}^{2} (\frac{d}{d x} [14 x] + \frac{d}{d x} [- x^{2}])$

Step 3

Evaluate $\frac{d}{d x} [14 x]$ .

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Step 3.1

Since $14$ is constant with respect to $x$ , the derivative of $14 x$ with respect to $x$ is $14 \frac{d}{d x} [x]$ .

$3 {(14 x - x^{2})}^{2} (14 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}])$

Step 3.2

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

$3 {(14 x - x^{2})}^{2} (14 \cdot 1 + \frac{d}{d x} [- x^{2}])$

Step 3.3

Multiply $14$ by $1$ .

$3 {(14 x - x^{2})}^{2} (14 + \frac{d}{d x} [- x^{2}])$

Step 4

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$3 {(14 x - x^{2})}^{2} (14 - \frac{d}{d x} [x^{2}])$

Step 4.2

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

$3 {(14 x - x^{2})}^{2} (14 - (2 x))$

Step 4.3

Multiply $2$ by $- 1$ .

$3 {(14 x - x^{2})}^{2} (14 - 2 x)$

Step 5

Reorder terms.

$3 {(14 x - x^{2})}^{2} (- 2 x + 14)$