Find dy/dx y=1/(x^12)

$y = \frac{1}{x^{12}}$

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1}{x^{12}})$

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

$y'$

Differentiate the right side of the equation.

Apply basic rules of exponents.

Rewrite $\frac{1}{x^{12}}$ as ${(x^{12})}^{- 1}$ .

$\frac{d}{d x} [{(x^{12})}^{- 1}]$

Multiply the exponents in ${(x^{12})}^{- 1}$ .

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

$\frac{d}{d x} [x^{12 \cdot - 1}]$

Multiply $12$ by $- 1$ .

$\frac{d}{d x} [x^{- 12}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 12$ .

$- 12 x^{- 13}$

Simplify.

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

$- 12 \frac{1}{x^{13}}$

Combine terms.

Combine $- 12$ and $\frac{1}{x^{13}}$ .

$\frac{- 12}{x^{13}}$

Move the negative in front of the fraction.

$- \frac{12}{x^{13}}$

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

$y' = - \frac{12}{x^{13}}$

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

$\frac{d y}{d x} = - \frac{12}{x^{13}}$