Calculus Examples

$y = 9 x^{- 5}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (9 x^{- 5})$

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

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{- 5}$ with respect to $x$ is $9 \frac{d}{d x} [x^{- 5}]$ .

$9 \frac{d}{d x} [x^{- 5}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 5$ .

$9 (- 5 x^{- 6})$

Step 3.3

Multiply $- 5$ by $9$ .

$- 45 x^{- 6}$

Step 3.4

Simplify.

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Step 3.4.1

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

$- 45 \frac{1}{x^{6}}$

Step 3.4.2

Combine $- 45$ and $\frac{1}{x^{6}}$ .

$\frac{- 45}{x^{6}}$

Step 3.4.3

Move the negative in front of the fraction.

$- \frac{45}{x^{6}}$

Step 4

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

$y' = - \frac{45}{x^{6}}$

Step 5

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

$\frac{d y}{d x} = - \frac{45}{x^{6}}$