Calculus Examples

$Y = x^{- 2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (Y) = \frac{d}{d x} (x^{- 2})$

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

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 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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 $\frac{d Y}{d x}$ .

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