Calculus Examples

$h (x) = 7 x^{2} {(3 x + 8)}^{4}$

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 $f (x) = 7 x^{2}$ and $g (x) = {(3 x + 8)}^{4}$ .

$7 x^{2} \frac{d}{d x} [{(3 x + 8)}^{4}] + {(3 x + 8)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 3 x + 8$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x + 8$ .

$7 x^{2} (\frac{d}{d u} [u^{4}] \frac{d}{d x} [3 x + 8]) + {(3 x + 8)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$7 x^{2} (4 u^{3} \frac{d}{d x} [3 x + 8]) + {(3 x + 8)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 2.3

Replace all occurrences of $u$ with $3 x + 8$ .

$7 x^{2} (4 {(3 x + 8)}^{3} \frac{d}{d x} [3 x + 8]) + {(3 x + 8)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $3 x + 8$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [8]$ .

$7 x^{2} (4 {(3 x + 8)}^{3} (\frac{d}{d x} [3 x] + \frac{d}{d x} [8])) + {(3 x + 8)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$7 x^{2} (4 {(3 x + 8)}^{3} (3 \frac{d}{d x} [x] + \frac{d}{d x} [8])) + {(3 x + 8)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.3

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

$7 x^{2} (4 {(3 x + 8)}^{3} (3 \cdot 1 + \frac{d}{d x} [8])) + {(3 x + 8)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.4

Multiply $3$ by $1$ .

$7 x^{2} (4 {(3 x + 8)}^{3} (3 + \frac{d}{d x} [8])) + {(3 x + 8)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.5

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$7 x^{2} (4 {(3 x + 8)}^{3} (3 + 0)) + {(3 x + 8)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $3$ and $0$ .

$7 x^{2} (4 {(3 x + 8)}^{3} \cdot 3) + {(3 x + 8)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.6.2

Multiply $3$ by $4$ .

$7 x^{2} (12 {(3 x + 8)}^{3}) + {(3 x + 8)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.7

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$7 x^{2} (12 {(3 x + 8)}^{3}) + {(3 x + 8)}^{4} (7 \frac{d}{d x} [x^{2}])$

Step 3.8

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

$7 x^{2} (12 {(3 x + 8)}^{3}) + {(3 x + 8)}^{4} (7 (2 x))$

Step 3.9

Multiply $2$ by $7$ .

$7 x^{2} (12 {(3 x + 8)}^{3}) + {(3 x + 8)}^{4} (14 x)$

Step 4

Simplify.

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Step 4.1

Multiply $12$ by $7$ .

$84 x^{2} {(3 x + 8)}^{3} + {(3 x + 8)}^{4} (14 x)$

Step 4.2

Move $14$ to the left of ${(3 x + 8)}^{4}$ .

$84 x^{2} {(3 x + 8)}^{3} + 14 \cdot {(3 x + 8)}^{4} x$

Step 4.3

Factor $14 x {(3 x + 8)}^{3}$ out of $84 x^{2} {(3 x + 8)}^{3} + 14 {(3 x + 8)}^{4} x$ .

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Step 4.3.1

Factor $14 x {(3 x + 8)}^{3}$ out of $84 x^{2} {(3 x + 8)}^{3}$ .

$14 x {(3 x + 8)}^{3} (6 x) + 14 {(3 x + 8)}^{4} x$

Step 4.3.2

Factor $14 x {(3 x + 8)}^{3}$ out of $14 {(3 x + 8)}^{4} x$ .

$14 x {(3 x + 8)}^{3} (6 x) + 14 x {(3 x + 8)}^{3} (3 x + 8)$

Step 4.3.3

Factor $14 x {(3 x + 8)}^{3}$ out of $14 x {(3 x + 8)}^{3} (6 x) + 14 x {(3 x + 8)}^{3} (3 x + 8)$ .

$14 x {(3 x + 8)}^{3} (6 x + 3 x + 8)$

Step 4.4

Add $6 x$ and $3 x$ .

$14 x {(3 x + 8)}^{3} (9 x + 8)$