Calculus Examples

$y = \frac{2 x}{3} + \frac{3 x}{2}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\frac{2 x}{3} + \frac{3 x}{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{2 x}{3}] + \frac{d}{d x} [\frac{3 x}{2}]$ .

$\frac{d}{d x} [\frac{2 x}{3}] + \frac{d}{d x} [\frac{3 x}{2}]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{2 x}{3}]$ .

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Step 1.2.1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2 x}{3}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [x]$ .

$\frac{2}{3} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{3 x}{2}]$

Step 1.2.2

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

$\frac{2}{3} \cdot 1 + \frac{d}{d x} [\frac{3 x}{2}]$

Step 1.2.3

Multiply $\frac{2}{3}$ by $1$ .

$\frac{2}{3} + \frac{d}{d x} [\frac{3 x}{2}]$

Step 1.3

Evaluate $\frac{d}{d x} [\frac{3 x}{2}]$ .

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Step 1.3.1

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 x}{2}$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [x]$ .

$\frac{2}{3} + \frac{3}{2} \frac{d}{d x} [x]$

Step 1.3.2

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

$\frac{2}{3} + \frac{3}{2} \cdot 1$

Step 1.3.3

Multiply $\frac{3}{2}$ by $1$ .

$\frac{2}{3} + \frac{3}{2}$

Step 1.4

Combine terms.

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Step 1.4.1

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2}{3} \cdot \frac{2}{2} + \frac{3}{2}$

Step 1.4.2

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} \cdot \frac{2}{2} + \frac{3}{2} \cdot \frac{3}{3}$

Step 1.4.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.3.1

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

$\frac{2 \cdot 2}{3 \cdot 2} + \frac{3}{2} \cdot \frac{3}{3}$

Step 1.4.3.2

Multiply $3$ by $2$ .

$\frac{2 \cdot 2}{6} + \frac{3}{2} \cdot \frac{3}{3}$

Step 1.4.3.3

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$\frac{2 \cdot 2}{6} + \frac{3 \cdot 3}{2 \cdot 3}$

Step 1.4.3.4

Multiply $2$ by $3$ .

$\frac{2 \cdot 2}{6} + \frac{3 \cdot 3}{6}$

Step 1.4.4

Combine the numerators over the common denominator.

$\frac{2 \cdot 2 + 3 \cdot 3}{6}$

Step 1.4.5

Simplify the numerator.

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Step 1.4.5.1

Multiply $2$ by $2$ .

$\frac{4 + 3 \cdot 3}{6}$

Step 1.4.5.2

Multiply $3$ by $3$ .

$\frac{4 + 9}{6}$

Step 1.4.5.3

Add $4$ and $9$ .

$f' (x) = \frac{13}{6}$

Step 2

Since $\frac{13}{6}$ is constant with respect to $x$ , the derivative of $\frac{13}{6}$ with respect to $x$ is $0$ .

$f'' (x) = 0$