Calculus Examples

$y = u^{\frac{3}{2}}$ , $u = 6 x + 5$

Step 1

The chain rule states that the derivative of $y$ with respect to $x$ is equal to the derivative of $y$ with respect to $u$ times the derivative of $u$ with respect to $x$ .

$\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$

Step 2

Find $\frac{d y}{d u}$ .

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Step 2.1

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

$\frac{3}{2} u^{\frac{3}{2} - 1}$

Step 2.2

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

$\frac{3}{2} u^{\frac{3}{2} - 1 \cdot \frac{2}{2}}$

Step 2.3

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

$\frac{3}{2} u^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3}{2} u^{\frac{3 - 1 \cdot 2}{2}}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 1$ by $2$ .

$\frac{3}{2} u^{\frac{3 - 2}{2}}$

Step 2.5.2

Subtract $2$ from $3$ .

$\frac{3}{2} u^{\frac{1}{2}}$

Step 2.6

Combine $\frac{3}{2}$ and $u^{\frac{1}{2}}$ .

$\frac{3 u^{\frac{1}{2}}}{2}$

Step 3

Find $\frac{d u}{d x}$ .

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Step 3.1

By the Sum Rule, the derivative of $6 x + 5$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [5]$

Step 3.2

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 3.2.1

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

$6 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 3.2.2

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

$6 \cdot 1 + \frac{d}{d x} [5]$

Step 3.2.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [5]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

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

$6 + 0$

Step 3.3.2

Add $6$ and $0$ .

$6$

Step 4

Multiply $\frac{d y}{d u}$ by $\frac{d u}{d x}$ .

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

Step 5

Simplify the right side $(6) \cdot (\frac{3 u^{\frac{1}{2}}}{2})$ .

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Step 5.1

Cancel the common factor of $2$ .

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Step 5.1.1

Factor $2$ out of $6$ .

$\frac{d y}{d x} = 2 (3) \cdot \frac{3 u^{\frac{1}{2}}}{2}$

Step 5.1.2

Cancel the common factor.

$\frac{d y}{d x} = 2 \cdot 3 \cdot \frac{3 u^{\frac{1}{2}}}{2}$

Step 5.1.3

Rewrite the expression.

$\frac{d y}{d x} = 3 \cdot (3 u^{\frac{1}{2}})$

Step 5.2

Multiply $3$ by $3$ .

$\frac{d y}{d x} = 9 u^{\frac{1}{2}}$

Step 6

Substitute in the value of $u$ into the derivative $9 u^{\frac{1}{2}}$ .

$\frac{d y}{d x} = 9 {(6 x + 5)}^{\frac{1}{2}}$