Calculus Examples

$y = {(8 x^{3} + 7)}^{\frac{3}{2}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = 8 x^{3} + 7$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $8 x^{3} + 7$ .

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{3}{2} u^{\frac{3}{2} - 1} \frac{d}{d x} [8 x^{3} + 7]$

Step 1.3

Replace all occurrences of $u$ with $8 x^{3} + 7$ .

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

Step 2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $2$ .

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

Step 5.2

Subtract $2$ from $3$ .

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

Step 6

Combine $\frac{3}{2}$ and ${(8 x^{3} + 7)}^{\frac{1}{2}}$ .

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

Step 7

By the Sum Rule, the derivative of $8 x^{3} + 7$ with respect to $x$ is $\frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [7]$ .

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

Step 8

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{3}$ with respect to $x$ is $8 \frac{d}{d x} [x^{3}]$ .

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

Step 9

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

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

Step 10

Multiply $3$ by $8$ .

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

Step 11

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{3 {(8 x^{3} + 7)}^{\frac{1}{2}}}{2} (24 x^{2} + 0)$

Step 12

Simplify terms.

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Step 12.1

Add $24 x^{2}$ and $0$ .

$\frac{3 {(8 x^{3} + 7)}^{\frac{1}{2}}}{2} (24 x^{2})$

Step 12.2

Combine $24$ and $\frac{3 {(8 x^{3} + 7)}^{\frac{1}{2}}}{2}$ .

$\frac{24 (3 {(8 x^{3} + 7)}^{\frac{1}{2}})}{2} x^{2}$

Step 12.3

Multiply $3$ by $24$ .

$\frac{72 {(8 x^{3} + 7)}^{\frac{1}{2}}}{2} x^{2}$

Step 12.4

Combine $\frac{72 {(8 x^{3} + 7)}^{\frac{1}{2}}}{2}$ and $x^{2}$ .

$\frac{72 {(8 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{2}$

Step 12.5

Factor $2$ out of $72 {(8 x^{3} + 7)}^{\frac{1}{2}} x^{2}$ .

$\frac{2 (36 {(8 x^{3} + 7)}^{\frac{1}{2}} x^{2})}{2}$

Step 13

Cancel the common factors.

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Step 13.1

Factor $2$ out of $2$ .

$\frac{2 (36 {(8 x^{3} + 7)}^{\frac{1}{2}} x^{2})}{2 (1)}$

Step 13.2

Cancel the common factor.

$\frac{2 (36 {(8 x^{3} + 7)}^{\frac{1}{2}} x^{2})}{2 \cdot 1}$

Step 13.3

Rewrite the expression.

$\frac{36 {(8 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{1}$

Step 13.4

Divide $36 {(8 x^{3} + 7)}^{\frac{1}{2}} x^{2}$ by $1$ .

$36 {(8 x^{3} + 7)}^{\frac{1}{2}} x^{2}$

Step 14

Reorder the factors of $36 {(8 x^{3} + 7)}^{\frac{1}{2}} x^{2}$ .

$36 x^{2} {(8 x^{3} + 7)}^{\frac{1}{2}}$