Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Divide $x$ by $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

$x - 1 \cdot 1$

Step 3.3

Multiply $- 1$ by $1$ .

$x - 1$