Find dy/dx y=1/2x^2-x

$y = \frac{1}{2} x^{2} - x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1}{2} x^{2} - x)$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

By the Sum Rule, the derivative of $\frac{1}{2} x^{2} - x$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [- x]$

Evaluate $\frac{d}{d x} [\frac{1}{2} x^{2}]$ .

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} x^{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{2}]$ .

$\frac{1}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{2} (2 x) + \frac{d}{d x} [- x]$

Combine $2$ and $\frac{1}{2}$ .

$\frac{2}{2} x + \frac{d}{d x} [- x]$

Combine $\frac{2}{2}$ and $x$ .

$\frac{2 x}{2} + \frac{d}{d x} [- x]$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} + \frac{d}{d x} [- x]$

Divide $x$ by $1$ .

$x + \frac{d}{d x} [- x]$

Evaluate $\frac{d}{d x} [- x]$ .

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

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

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

$x - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$x - 1$

Step 4

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

$y' = x - 1$

Step 5

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

$\frac{d y}{d x} = x - 1$

Find dy/dx y=1/2x^2-x

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