Calculus Examples

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

Step 1

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

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

Step 2.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 2.2.1

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

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

Step 2.2.2

Multiply $3$ by $- 1$ .

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

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Combine terms.

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Step 4.2.1

Combine $- 3$ and $\frac{1}{x^{4}}$ .

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

Step 4.2.2

Move the negative in front of the fraction.

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

Step 4.2.3

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

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

Step 4.2.4

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

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