Calculus Examples

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

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

Step 2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 3

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

$- \frac{1}{2} (- 2 x^{- 3})$

Step 4

Simplify terms.

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Step 4.1

Multiply $- 2$ by $- 1$ .

$2 (\frac{1}{2}) x^{- 3}$

Step 4.2

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

$\frac{2}{2} x^{- 3}$

Step 4.3

Combine $\frac{2}{2}$ and $x^{- 3}$ .

$\frac{2 x^{- 3}}{2}$

Step 4.4

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{2 x^{3}}$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Cancel the common factor.

$\frac{2}{2 x^{3}}$

Step 4.5.2

Rewrite the expression.

$\frac{1}{x^{3}}$