Calculus Examples

$f (x) = x^{7} {(5 + 8 x)}^{3}$

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 $g (x) = {(5 + 8 x)}^{3}$ .

$x^{7} \frac{d}{d x} [{(5 + 8 x)}^{3}] + {(5 + 8 x)}^{3} \frac{d}{d x} [x^{7}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 5 + 8 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $5 + 8 x$ .

$x^{7} (\frac{d}{d u} [u^{3}] \frac{d}{d x} [5 + 8 x]) + {(5 + 8 x)}^{3} \frac{d}{d x} [x^{7}]$

Step 2.2

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

$x^{7} (3 u^{2} \frac{d}{d x} [5 + 8 x]) + {(5 + 8 x)}^{3} \frac{d}{d x} [x^{7}]$

Step 2.3

Replace all occurrences of $u$ with $5 + 8 x$ .

$x^{7} (3 {(5 + 8 x)}^{2} \frac{d}{d x} [5 + 8 x]) + {(5 + 8 x)}^{3} \frac{d}{d x} [x^{7}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $5 + 8 x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [8 x]$ .

$x^{7} (3 {(5 + 8 x)}^{2} (\frac{d}{d x} [5] + \frac{d}{d x} [8 x])) + {(5 + 8 x)}^{3} \frac{d}{d x} [x^{7}]$

Step 3.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$x^{7} (3 {(5 + 8 x)}^{2} (0 + \frac{d}{d x} [8 x])) + {(5 + 8 x)}^{3} \frac{d}{d x} [x^{7}]$

Step 3.3

Add $0$ and $\frac{d}{d x} [8 x]$ .

$x^{7} (3 {(5 + 8 x)}^{2} \frac{d}{d x} [8 x]) + {(5 + 8 x)}^{3} \frac{d}{d x} [x^{7}]$

Step 3.4

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$x^{7} (3 {(5 + 8 x)}^{2} (8 \frac{d}{d x} [x])) + {(5 + 8 x)}^{3} \frac{d}{d x} [x^{7}]$

Step 3.5

Multiply $8$ by $3$ .

$x^{7} (24 {(5 + 8 x)}^{2} \frac{d}{d x} [x]) + {(5 + 8 x)}^{3} \frac{d}{d x} [x^{7}]$

Step 3.6

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

$x^{7} (24 {(5 + 8 x)}^{2} \cdot 1) + {(5 + 8 x)}^{3} \frac{d}{d x} [x^{7}]$

Step 3.7

Multiply $24$ by $1$ .

$x^{7} (24 {(5 + 8 x)}^{2}) + {(5 + 8 x)}^{3} \frac{d}{d x} [x^{7}]$

Step 3.8

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

$x^{7} (24 {(5 + 8 x)}^{2}) + {(5 + 8 x)}^{3} (7 x^{6})$

Step 3.9

Move $7$ to the left of ${(5 + 8 x)}^{3}$ .

$x^{7} (24 {(5 + 8 x)}^{2}) + 7 \cdot {(5 + 8 x)}^{3} x^{6}$

Step 4

Simplify.

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Step 4.1

Factor $x^{6} {(5 + 8 x)}^{2}$ out of $x^{7} (24 {(5 + 8 x)}^{2}) + 7 {(5 + 8 x)}^{3} x^{6}$ .

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Step 4.1.1

Factor $x^{6} {(5 + 8 x)}^{2}$ out of $x^{7} (24 {(5 + 8 x)}^{2})$ .

$x^{6} {(5 + 8 x)}^{2} (x \cdot (24)) + 7 {(5 + 8 x)}^{3} x^{6}$

Step 4.1.2

Factor $x^{6} {(5 + 8 x)}^{2}$ out of $7 {(5 + 8 x)}^{3} x^{6}$ .

$x^{6} {(5 + 8 x)}^{2} (x \cdot (24)) + x^{6} {(5 + 8 x)}^{2} (7 (5 + 8 x))$

Step 4.1.3

Factor $x^{6} {(5 + 8 x)}^{2}$ out of $x^{6} {(5 + 8 x)}^{2} (x \cdot (24)) + x^{6} {(5 + 8 x)}^{2} (7 (5 + 8 x))$ .

$x^{6} {(5 + 8 x)}^{2} (x \cdot (24) + 7 (5 + 8 x))$

Step 4.2

Move $24$ to the left of $x$ .

$x^{6} {(5 + 8 x)}^{2} (24 x + 7 (5 + 8 x))$