Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1}{x^{2}})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Apply basic rules of exponents.

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Step 3.1.1

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

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

Multiply $2$ by $- 1$ .

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

Step 3.2

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

$- 2 x^{- 3}$

Step 3.3

Simplify.

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Step 3.3.1

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

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

Step 3.3.2

Combine terms.

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Step 3.3.2.1

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

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

Step 3.3.2.2

Move the negative in front of the fraction.

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{2}{x^{3}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

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