Calculus Examples

$y = \frac{1}{27} {(9 x^{2} + 6)}^{\frac{3}{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1}{27} \cdot {(9 x^{2} + 6)}^{\frac{3}{2}})$

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

Since $\frac{1}{27}$ is constant with respect to $x$ , the derivative of $\frac{1}{27} \cdot {(9 x^{2} + 6)}^{\frac{3}{2}}$ with respect to $x$ is $\frac{1}{27} \frac{d}{d x} [{(9 x^{2} + 6)}^{\frac{3}{2}}]$ .

$\frac{1}{27} \frac{d}{d x} [{(9 x^{2} + 6)}^{\frac{3}{2}}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = 9 x^{2} + 6$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $9 x^{2} + 6$ .

$\frac{1}{27} (\frac{d}{d u} [u^{\frac{3}{2}}] \frac{d}{d x} [9 x^{2} + 6])$

Step 3.2.2

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

$\frac{1}{27} (\frac{3}{2} u^{\frac{3}{2} - 1} \frac{d}{d x} [9 x^{2} + 6])$

Step 3.2.3

Replace all occurrences of $u$ with $9 x^{2} + 6$ .

$\frac{1}{27} (\frac{3}{2} {(9 x^{2} + 6)}^{\frac{3}{2} - 1} \frac{d}{d x} [9 x^{2} + 6])$

Step 3.3

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

$\frac{1}{27} (\frac{3}{2} {(9 x^{2} + 6)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [9 x^{2} + 6])$

Step 3.4

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

$\frac{1}{27} (\frac{3}{2} {(9 x^{2} + 6)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [9 x^{2} + 6])$

Step 3.5

Combine the numerators over the common denominator.

$\frac{1}{27} (\frac{3}{2} {(9 x^{2} + 6)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} [9 x^{2} + 6])$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{27} (\frac{3}{2} {(9 x^{2} + 6)}^{\frac{3 - 2}{2}} \frac{d}{d x} [9 x^{2} + 6])$

Step 3.6.2

Subtract $2$ from $3$ .

$\frac{1}{27} (\frac{3}{2} {(9 x^{2} + 6)}^{\frac{1}{2}} \frac{d}{d x} [9 x^{2} + 6])$

Step 3.7

Combine $\frac{3}{2}$ and ${(9 x^{2} + 6)}^{\frac{1}{2}}$ .

$\frac{1}{27} (\frac{3 {(9 x^{2} + 6)}^{\frac{1}{2}}}{2} \frac{d}{d x} [9 x^{2} + 6])$

Step 3.8

Multiply $\frac{3 {(9 x^{2} + 6)}^{\frac{1}{2}}}{2}$ by $\frac{1}{27}$ .

$\frac{3 {(9 x^{2} + 6)}^{\frac{1}{2}}}{2 \cdot 27} \frac{d}{d x} [9 x^{2} + 6]$

Step 3.9

Multiply $2$ by $27$ .

$\frac{3 {(9 x^{2} + 6)}^{\frac{1}{2}}}{54} \frac{d}{d x} [9 x^{2} + 6]$

Step 3.10

Factor $3$ out of $3 {(9 x^{2} + 6)}^{\frac{1}{2}}$ .

$\frac{3 ({(9 x^{2} + 6)}^{\frac{1}{2}})}{54} \frac{d}{d x} [9 x^{2} + 6]$

Step 3.11

Cancel the common factors.

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Step 3.11.1

Factor $3$ out of $54$ .

$\frac{3 {(9 x^{2} + 6)}^{\frac{1}{2}}}{3 \cdot 18} \frac{d}{d x} [9 x^{2} + 6]$

Step 3.11.2

Cancel the common factor.

$\frac{3 {(9 x^{2} + 6)}^{\frac{1}{2}}}{3 \cdot 18} \frac{d}{d x} [9 x^{2} + 6]$

Step 3.11.3

Rewrite the expression.

$\frac{{(9 x^{2} + 6)}^{\frac{1}{2}}}{18} \frac{d}{d x} [9 x^{2} + 6]$

Step 3.12

By the Sum Rule, the derivative of $9 x^{2} + 6$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [6]$ .

$\frac{{(9 x^{2} + 6)}^{\frac{1}{2}}}{18} (\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [6])$

Step 3.13

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$\frac{{(9 x^{2} + 6)}^{\frac{1}{2}}}{18} (9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6])$

Step 3.14

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

$\frac{{(9 x^{2} + 6)}^{\frac{1}{2}}}{18} (9 (2 x) + \frac{d}{d x} [6])$

Step 3.15

Multiply $2$ by $9$ .

$\frac{{(9 x^{2} + 6)}^{\frac{1}{2}}}{18} (18 x + \frac{d}{d x} [6])$

Step 3.16

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{{(9 x^{2} + 6)}^{\frac{1}{2}}}{18} (18 x + 0)$

Step 3.17

Simplify terms.

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Step 3.17.1

Add $18 x$ and $0$ .

$\frac{{(9 x^{2} + 6)}^{\frac{1}{2}}}{18} (18 x)$

Step 3.17.2

Combine $18$ and $\frac{{(9 x^{2} + 6)}^{\frac{1}{2}}}{18}$ .

$\frac{18 {(9 x^{2} + 6)}^{\frac{1}{2}}}{18} x$

Step 3.17.3

Combine $\frac{18 {(9 x^{2} + 6)}^{\frac{1}{2}}}{18}$ and $x$ .

$\frac{18 {(9 x^{2} + 6)}^{\frac{1}{2}} x}{18}$

Step 3.17.4

Cancel the common factor.

$\frac{18 {(9 x^{2} + 6)}^{\frac{1}{2}} x}{18}$

Step 3.17.5

Simplify the expression.

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Step 3.17.5.1

Divide ${(9 x^{2} + 6)}^{\frac{1}{2}} x$ by $1$ .

${(9 x^{2} + 6)}^{\frac{1}{2}} x$

Step 3.17.5.2

Reorder the factors of ${(9 x^{2} + 6)}^{\frac{1}{2}} x$ .

$x {(9 x^{2} + 6)}^{\frac{1}{2}}$

Step 4

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

$y' = x {(9 x^{2} + 6)}^{\frac{1}{2}}$

Step 5

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

$\frac{d y}{d x} = x {(9 x^{2} + 6)}^{\frac{1}{2}}$