Calculus Examples

$- \frac{3}{x^{4} y^{2}}$

Step 1

Since $- \frac{3}{x^{4}}$ is constant with respect to $y$ , the derivative of $- \frac{3}{x^{4} y^{2}}$ with respect to $y$ is $- \frac{3}{x^{4}} \frac{d}{d y} [\frac{1}{y^{2}}]$ .

$- \frac{3}{x^{4}} \frac{d}{d y} [\frac{1}{y^{2}}]$

Step 2

Apply basic rules of exponents.

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Step 2.1

Rewrite $\frac{1}{y^{2}}$ as ${(y^{2})}^{- 1}$ .

$- \frac{3}{x^{4}} \frac{d}{d y} [{(y^{2})}^{- 1}]$

Step 2.2

Multiply the exponents in ${(y^{2})}^{- 1}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{3}{x^{4}} \frac{d}{d y} [y^{2 \cdot - 1}]$

Step 2.2.2

Multiply $2$ by $- 1$ .

$- \frac{3}{x^{4}} \frac{d}{d y} [y^{- 2}]$

Step 3

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

$- \frac{3}{x^{4}} (- 2 y^{- 3})$

Step 4

Combine fractions.

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Step 4.1

Multiply $- 2$ by $- 1$ .

$2 \frac{3}{x^{4}} y^{- 3}$

Step 4.2

Combine $2$ and $\frac{3}{x^{4}}$ .

$\frac{2 \cdot 3}{x^{4}} y^{- 3}$

Step 4.3

Multiply $2$ by $3$ .

$\frac{6}{x^{4}} y^{- 3}$

Step 4.4

Combine $\frac{6}{x^{4}}$ and $y^{- 3}$ .

$\frac{6 y^{- 3}}{x^{4}}$

Step 4.5

Move $y^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{6}{x^{4} y^{3}}$