Calculus Examples

$- \frac{5 x^{2}}{4 y^{2}}$

Step 1

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

Step 2

Since $- \frac{5 x^{2}}{4}$ is constant with respect to $y$ , the derivative of $- \frac{5 x^{2}}{4 y^{2}}$ with respect to $y$ is $- \frac{5 x^{2}}{4} \frac{d}{d y} [\frac{1}{y^{2}}]$ .

$- \frac{5 x^{2}}{4} \frac{d}{d y} [\frac{1}{y^{2}}]$

Step 3

Apply basic rules of exponents.

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Step 3.1

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

$- \frac{5 x^{2}}{4} \frac{d}{d y} [{(y^{2})}^{- 1}]$

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, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{5 x^{2}}{4} \frac{d}{d y} [y^{2 \cdot - 1}]$

Step 3.2.2

Multiply $2$ by $- 1$ .

$- \frac{5 x^{2}}{4} \frac{d}{d y} [y^{- 2}]$

Step 4

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

$- \frac{5 x^{2}}{4} (- 2 y^{- 3})$

Step 5

Simplify terms.

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Step 5.1

Multiply $- 2$ by $- 1$ .

$2 \frac{5 x^{2}}{4} y^{- 3}$

Step 5.2

Combine $2$ and $\frac{5 x^{2}}{4}$ .

$\frac{2 (5 x^{2})}{4} y^{- 3}$

Step 5.3

Multiply $5$ by $2$ .

$\frac{10 x^{2}}{4} y^{- 3}$

Step 5.4

Combine $\frac{10 x^{2}}{4}$ and $y^{- 3}$ .

$\frac{10 x^{2} y^{- 3}}{4}$

Step 5.5

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

$\frac{10 x^{2}}{4 y^{3}}$

Step 5.6

Cancel the common factor of $10$ and $4$ .

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Step 5.6.1

Factor $2$ out of $10 x^{2}$ .

$\frac{2 (5 x^{2})}{4 y^{3}}$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Factor $2$ out of $4 y^{3}$ .

$\frac{2 (5 x^{2})}{2 (2 y^{3})}$

Step 5.6.2.2

Cancel the common factor.

$\frac{2 (5 x^{2})}{2 (2 y^{3})}$

Step 5.6.2.3

Rewrite the expression.

$\frac{5 x^{2}}{2 y^{3}}$