Find dy/dx y=1/(x^7)

$y = \frac{1}{x^{7}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1}{x^{7}})$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

Apply basic rules of exponents.

Rewrite $\frac{1}{x^{7}}$ as ${(x^{7})}^{- 1}$ .

$\frac{d}{d x} [{(x^{7})}^{- 1}]$

Multiply the exponents in ${(x^{7})}^{- 1}$ .

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

$\frac{d}{d x} [x^{7 \cdot - 1}]$

Multiply $7$ by $- 1$ .

$\frac{d}{d x} [x^{- 7}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 7$ .

$- 7 x^{- 8}$

Simplify.

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

$- 7 \frac{1}{x^{8}}$

Combine terms.

Combine $- 7$ and $\frac{1}{x^{8}}$ .

$\frac{- 7}{x^{8}}$

Move the negative in front of the fraction.

$- \frac{7}{x^{8}}$

Step 4

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

$y' = - \frac{7}{x^{8}}$

Step 5

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

$\frac{d y}{d x} = - \frac{7}{x^{8}}$