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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Move $y^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $3$ .

$- \frac{x^{2}}{2 y^{2}} + \frac{6}{x} y$

Step 3.5

Combine $\frac{6}{x}$ and $y$ .

$- \frac{x^{2}}{2 y^{2}} + \frac{6 y}{x}$

Step 4

Reorder terms.

$\frac{6 y}{x} - \frac{x^{2}}{2 y^{2}}$