Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Since $\frac{1}{2 y}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{2 y}$ with respect to $x$ is $\frac{1}{2 y} \frac{d}{d x} [x^{2}]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine $\frac{2}{2 y}$ and $x$ .

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

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

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

Step 3

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

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Step 3.1

Since $3 y^{2}$ is constant with respect to $x$ , the derivative of $\frac{3 y^{2}}{x}$ with respect to $x$ is $3 y^{2} \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 3.2

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

$\frac{x}{y} + 3 y^{2} \frac{d}{d x} [x^{- 1}]$

Step 3.3

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

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

Step 3.4

Multiply $- 1$ by $3$ .

$\frac{x}{y} - 3 y^{2} x^{- 2}$

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Combine terms.

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Step 4.2.1

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

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

Step 4.2.2

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

$\frac{x}{y} + \frac{y^{2} \cdot - 3}{x^{2}}$

Step 4.2.3

Move $- 3$ to the left of $y^{2}$ .

$\frac{x}{y} + \frac{- 3 \cdot y^{2}}{x^{2}}$

Step 4.2.4

Move the negative in front of the fraction.

$\frac{x}{y} - \frac{3 y^{2}}{x^{2}}$