Calculus Examples

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

Step 1

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

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

Step 2

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

$- \frac{1}{2} (2 x)$

Step 3

Simplify terms.

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Step 3.1

Multiply $2$ by $- 1$ .

$- 2 (\frac{1}{2}) x$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Cancel the common factor of $- 2$ and $2$ .

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Step 3.4.1

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x)}{2}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $2$ .

$\frac{2 (- x)}{2 (1)}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 (- x)}{2 \cdot 1}$

Step 3.4.2.3

Rewrite the expression.

$\frac{- x}{1}$

Step 3.4.2.4

Divide $- x$ by $1$ .

$- x$