Calculus Examples

$y = {(6 x^{3} + 7)}^{\frac{3}{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(6 x^{3} + 7)}^{\frac{3}{2}})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $g (x) = 6 x^{3} + 7$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $6 x^{3} + 7$ .

$\frac{d}{d u} [u^{\frac{3}{2}}] \frac{d}{d x} [6 x^{3} + 7]$

Step 3.1.2

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

$\frac{3}{2} u^{\frac{3}{2} - 1} \frac{d}{d x} [6 x^{3} + 7]$

Step 3.1.3

Replace all occurrences of $u$ with $6 x^{3} + 7$ .

$\frac{3}{2} {(6 x^{3} + 7)}^{\frac{3}{2} - 1} \frac{d}{d x} [6 x^{3} + 7]$

Step 3.2

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

$\frac{3}{2} {(6 x^{3} + 7)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [6 x^{3} + 7]$

Step 3.3

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

$\frac{3}{2} {(6 x^{3} + 7)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [6 x^{3} + 7]$

Step 3.4

Combine the numerators over the common denominator.

$\frac{3}{2} {(6 x^{3} + 7)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} [6 x^{3} + 7]$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 1$ by $2$ .

$\frac{3}{2} {(6 x^{3} + 7)}^{\frac{3 - 2}{2}} \frac{d}{d x} [6 x^{3} + 7]$

Step 3.5.2

Subtract $2$ from $3$ .

$\frac{3}{2} {(6 x^{3} + 7)}^{\frac{1}{2}} \frac{d}{d x} [6 x^{3} + 7]$

Step 3.6

Combine $\frac{3}{2}$ and ${(6 x^{3} + 7)}^{\frac{1}{2}}$ .

$\frac{3 {(6 x^{3} + 7)}^{\frac{1}{2}}}{2} \frac{d}{d x} [6 x^{3} + 7]$

Step 3.7

By the Sum Rule, the derivative of $6 x^{3} + 7$ with respect to $x$ is $\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [7]$ .

$\frac{3 {(6 x^{3} + 7)}^{\frac{1}{2}}}{2} (\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [7])$

Step 3.8

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3}$ with respect to $x$ is $6 \frac{d}{d x} [x^{3}]$ .

$\frac{3 {(6 x^{3} + 7)}^{\frac{1}{2}}}{2} (6 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [7])$

Step 3.9

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

$\frac{3 {(6 x^{3} + 7)}^{\frac{1}{2}}}{2} (6 (3 x^{2}) + \frac{d}{d x} [7])$

Step 3.10

Multiply $3$ by $6$ .

$\frac{3 {(6 x^{3} + 7)}^{\frac{1}{2}}}{2} (18 x^{2} + \frac{d}{d x} [7])$

Step 3.11

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

$\frac{3 {(6 x^{3} + 7)}^{\frac{1}{2}}}{2} (18 x^{2} + 0)$

Step 3.12

Simplify terms.

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Step 3.12.1

Add $18 x^{2}$ and $0$ .

$\frac{3 {(6 x^{3} + 7)}^{\frac{1}{2}}}{2} (18 x^{2})$

Step 3.12.2

Combine $18$ and $\frac{3 {(6 x^{3} + 7)}^{\frac{1}{2}}}{2}$ .

$\frac{18 (3 {(6 x^{3} + 7)}^{\frac{1}{2}})}{2} x^{2}$

Step 3.12.3

Multiply $3$ by $18$ .

$\frac{54 {(6 x^{3} + 7)}^{\frac{1}{2}}}{2} x^{2}$

Step 3.12.4

Combine $\frac{54 {(6 x^{3} + 7)}^{\frac{1}{2}}}{2}$ and $x^{2}$ .

$\frac{54 {(6 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{2}$

Step 3.12.5

Factor $2$ out of $54 {(6 x^{3} + 7)}^{\frac{1}{2}} x^{2}$ .

$\frac{2 (27 {(6 x^{3} + 7)}^{\frac{1}{2}} x^{2})}{2}$

Step 3.13

Cancel the common factors.

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Step 3.13.1

Factor $2$ out of $2$ .

$\frac{2 (27 {(6 x^{3} + 7)}^{\frac{1}{2}} x^{2})}{2 (1)}$

Step 3.13.2

Cancel the common factor.

$\frac{2 (27 {(6 x^{3} + 7)}^{\frac{1}{2}} x^{2})}{2 \cdot 1}$

Step 3.13.3

Rewrite the expression.

$\frac{27 {(6 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{1}$

Step 3.13.4

Divide $27 {(6 x^{3} + 7)}^{\frac{1}{2}} x^{2}$ by $1$ .

$27 {(6 x^{3} + 7)}^{\frac{1}{2}} x^{2}$

Step 3.14

Reorder the factors of $27 {(6 x^{3} + 7)}^{\frac{1}{2}} x^{2}$ .

$27 x^{2} {(6 x^{3} + 7)}^{\frac{1}{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 27 x^{2} {(6 x^{3} + 7)}^{\frac{1}{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 27 x^{2} {(6 x^{3} + 7)}^{\frac{1}{2}}$