Calculus Examples

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

Step 1

This derivative could not be completed using the quotient rule. Mathway will use another method.

Step 2

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

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

Multiply $- 2$ by $- 1$ .

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Combine terms.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

Move $2$ to the left of $x^{2}$ .

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