Calculus Examples

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

Step 1

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

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

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

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 3

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

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

Step 4

Multiply $- 2$ by $2$ .

$- 4 x y^{- 3}$

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Combine terms.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

Move $- 4$ to the left of $x$ .

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

Step 5.2.4

Move the negative in front of the fraction.

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