Calculus Examples

${(4 x - x^{2})}^{3}$

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 $g (x) = 4 x - x^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $4 x - x^{2}$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [4 x - x^{2}]$

Step 1.2

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

$3 u^{2} \frac{d}{d x} [4 x - x^{2}]$

Step 1.3

Replace all occurrences of $u$ with $4 x - x^{2}$ .

$3 {(4 x - x^{2})}^{2} \frac{d}{d x} [4 x - x^{2}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- x^{2}]$ .

$3 {(4 x - x^{2})}^{2} (\frac{d}{d x} [4 x] + \frac{d}{d x} [- x^{2}])$

Step 2.2

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

$3 {(4 x - x^{2})}^{2} (4 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{2}])$

Step 2.3

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

$3 {(4 x - x^{2})}^{2} (4 \cdot 1 + \frac{d}{d x} [- x^{2}])$

Step 2.4

Multiply $4$ by $1$ .

$3 {(4 x - x^{2})}^{2} (4 + \frac{d}{d x} [- x^{2}])$

Step 2.5

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

$3 {(4 x - x^{2})}^{2} (4 - \frac{d}{d x} [x^{2}])$

Step 2.6

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

$3 {(4 x - x^{2})}^{2} (4 - (2 x))$

Step 2.7

Multiply $2$ by $- 1$ .

$3 {(4 x - x^{2})}^{2} (4 - 2 x)$