Calculus Examples

$y = \frac{x}{3} - \frac{3}{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x}{3} - \frac{3}{x})$

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

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

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

Step 3.2

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

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Step 3.2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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Step 3.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3}{x}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{1}{3} - 3 \frac{d}{d x} [\frac{1}{x}]$

Step 3.3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$\frac{1}{3} - 3 \frac{d}{d x} [x^{- 1}]$

Step 3.3.3

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

$\frac{1}{3} - 3 (- x^{- 2})$

Step 3.3.4

Multiply $- 1$ by $- 3$ .

$\frac{1}{3} + 3 x^{- 2}$

Step 3.4

Simplify.

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Step 3.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3} + 3 \frac{1}{x^{2}}$

Step 3.4.2

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

$\frac{1}{3} + \frac{3}{x^{2}}$

Step 3.4.3

Reorder terms.

$\frac{3}{x^{2}} + \frac{1}{3}$

Step 4

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

$y' = \frac{3}{x^{2}} + \frac{1}{3}$

Step 5

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

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