Find dy/dx y=5/x

$y = \frac{5}{x}$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x}]$ .

$5 \frac{d}{d x} [\frac{1}{x}]$

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

$5 \frac{d}{d x} [x^{- 1}]$

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

$5 (- x^{- 2})$

Multiply $- 1$ by $5$ .

$- 5 x^{- 2}$

Simplify.

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

$- 5 \frac{1}{x^{2}}$

Combine terms.

Combine $- 5$ and $\frac{1}{x^{2}}$ .

$\frac{- 5}{x^{2}}$

Move the negative in front of the fraction.

$- \frac{5}{x^{2}}$

Step 4

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

$y' = - \frac{5}{x^{2}}$

Step 5

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

$\frac{d y}{d x} = - \frac{5}{x^{2}}$