Calculus Examples

$f (x) = x^{7} {(4 - x)}^{8}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{7}$ and $g (x) = {(4 - x)}^{8}$ .

$x^{7} \frac{d}{d x} [{(4 - x)}^{8}] + {(4 - x)}^{8} \frac{d}{d x} [x^{7}]$

Step 2

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 $g (x) = 4 - x$ .

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Step 2.1

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

$x^{7} (\frac{d}{d u} [u^{8}] \frac{d}{d x} [4 - x]) + {(4 - x)}^{8} \frac{d}{d x} [x^{7}]$

Step 2.2

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

$x^{7} (8 u^{7} \frac{d}{d x} [4 - x]) + {(4 - x)}^{8} \frac{d}{d x} [x^{7}]$

Step 2.3

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

$x^{7} (8 {(4 - x)}^{7} \frac{d}{d x} [4 - x]) + {(4 - x)}^{8} \frac{d}{d x} [x^{7}]$

Step 3

Differentiate.

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Step 3.1

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

$x^{7} (8 {(4 - x)}^{7} (\frac{d}{d x} [4] + \frac{d}{d x} [- x])) + {(4 - x)}^{8} \frac{d}{d x} [x^{7}]$

Step 3.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$x^{7} (8 {(4 - x)}^{7} (0 + \frac{d}{d x} [- x])) + {(4 - x)}^{8} \frac{d}{d x} [x^{7}]$

Step 3.3

Add $0$ and $\frac{d}{d x} [- x]$ .

$x^{7} (8 {(4 - x)}^{7} \frac{d}{d x} [- x]) + {(4 - x)}^{8} \frac{d}{d x} [x^{7}]$

Step 3.4

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

$x^{7} (8 {(4 - x)}^{7} (- \frac{d}{d x} [x])) + {(4 - x)}^{8} \frac{d}{d x} [x^{7}]$

Step 3.5

Multiply $- 1$ by $8$ .

$x^{7} (- 8 {(4 - x)}^{7} \frac{d}{d x} [x]) + {(4 - x)}^{8} \frac{d}{d x} [x^{7}]$

Step 3.6

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

$x^{7} (- 8 {(4 - x)}^{7} \cdot 1) + {(4 - x)}^{8} \frac{d}{d x} [x^{7}]$

Step 3.7

Multiply $- 8$ by $1$ .

$x^{7} (- 8 {(4 - x)}^{7}) + {(4 - x)}^{8} \frac{d}{d x} [x^{7}]$

Step 3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$x^{7} (- 8 {(4 - x)}^{7}) + {(4 - x)}^{8} (7 x^{6})$

Step 3.9

Move $7$ to the left of ${(4 - x)}^{8}$ .

$x^{7} (- 8 {(4 - x)}^{7}) + 7 \cdot {(4 - x)}^{8} x^{6}$

Step 4

Simplify.

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Step 4.1

Factor $x^{6} {(4 - x)}^{7}$ out of $x^{7} (- 8 {(4 - x)}^{7}) + 7 {(4 - x)}^{8} x^{6}$ .

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Step 4.1.1

Factor $x^{6} {(4 - x)}^{7}$ out of $x^{7} (- 8 {(4 - x)}^{7})$ .

$x^{6} {(4 - x)}^{7} (x \cdot (- 8)) + 7 {(4 - x)}^{8} x^{6}$

Step 4.1.2

Factor $x^{6} {(4 - x)}^{7}$ out of $7 {(4 - x)}^{8} x^{6}$ .

$x^{6} {(4 - x)}^{7} (x \cdot (- 8)) + x^{6} {(4 - x)}^{7} (7 (4 - x))$

Step 4.1.3

Factor $x^{6} {(4 - x)}^{7}$ out of $x^{6} {(4 - x)}^{7} (x \cdot (- 8)) + x^{6} {(4 - x)}^{7} (7 (4 - x))$ .

$x^{6} {(4 - x)}^{7} (x \cdot (- 8) + 7 (4 - x))$

Step 4.2

Move $- 8$ to the left of $x$ .

$x^{6} {(4 - x)}^{7} (- 8 x + 7 (4 - x))$