Calculus Examples

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

Step 1

Cancel the common factor of $y$ and $y^{2}$ .

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Step 1.1

Raise $y$ to the power of $1$ .

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

Step 1.2

Factor $y$ out of $y^{1}$ .

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

Step 1.3

Cancel the common factors.

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Step 1.3.1

Factor $y$ out of $3 y^{2}$ .

$\frac{d}{d y} [\frac{y \cdot 1}{y (3 y)}]$

Step 1.3.2

Cancel the common factor.

$\frac{d}{d y} [\frac{y \cdot 1}{y (3 y)}]$

Step 1.3.3

Rewrite the expression.

$\frac{d}{d y} [\frac{1}{3 y}]$

Step 2

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

$\frac{1}{3} \frac{d}{d y} [\frac{1}{y}]$

Step 3

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

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

Step 4

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

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

Step 5

Combine fractions.

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Step 5.1

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

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

Step 5.2

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

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