Calculus Examples

$4 x^{- 2} + \frac{1}{4} x^{2} + 4$

Step 1

By the Sum Rule, the derivative of $4 x^{- 2} + \frac{1}{4} x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [4 x^{- 2}] + \frac{d}{d x} [\frac{1}{4} x^{2}] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [4 x^{- 2}] + \frac{d}{d x} [\frac{1}{4} x^{2}] + \frac{d}{d x} [4]$

Step 2

Evaluate $\frac{d}{d x} [4 x^{- 2}]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{- 2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{- 2}]$ .

$4 \frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [\frac{1}{4} x^{2}] + \frac{d}{d x} [4]$

Step 2.2

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

$4 (- 2 x^{- 3}) + \frac{d}{d x} [\frac{1}{4} x^{2}] + \frac{d}{d x} [4]$

Step 2.3

Multiply $- 2$ by $4$ .

$- 8 x^{- 3} + \frac{d}{d x} [\frac{1}{4} x^{2}] + \frac{d}{d x} [4]$

Step 3

Evaluate $\frac{d}{d x} [\frac{1}{4} x^{2}]$ .

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Step 3.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} x^{2}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{2}]$ .

$- 8 x^{- 3} + \frac{1}{4} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$

Step 3.2

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

$- 8 x^{- 3} + \frac{1}{4} (2 x) + \frac{d}{d x} [4]$

Step 3.3

Combine $2$ and $\frac{1}{4}$ .

$- 8 x^{- 3} + \frac{2}{4} x + \frac{d}{d x} [4]$

Step 3.4

Combine $\frac{2}{4}$ and $x$ .

$- 8 x^{- 3} + \frac{2 x}{4} + \frac{d}{d x} [4]$

Step 3.5

Cancel the common factor of $2$ and $4$ .

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Step 3.5.1

Factor $2$ out of $2 x$ .

$- 8 x^{- 3} + \frac{2 (x)}{4} + \frac{d}{d x} [4]$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $2$ out of $4$ .

$- 8 x^{- 3} + \frac{2 x}{2 \cdot 2} + \frac{d}{d x} [4]$

Step 3.5.2.2

Cancel the common factor.

$- 8 x^{- 3} + \frac{2 x}{2 \cdot 2} + \frac{d}{d x} [4]$

Step 3.5.2.3

Rewrite the expression.

$- 8 x^{- 3} + \frac{x}{2} + \frac{d}{d x} [4]$

Step 4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$- 8 x^{- 3} + \frac{x}{2} + 0$

Step 5

Simplify.

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Step 5.1

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

$- 8 \frac{1}{x^{3}} + \frac{x}{2} + 0$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $- 8$ and $\frac{1}{x^{3}}$ .

$\frac{- 8}{x^{3}} + \frac{x}{2} + 0$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{8}{x^{3}} + \frac{x}{2} + 0$

Step 5.2.3

Add $- \frac{8}{x^{3}} + \frac{x}{2}$ and $0$ .

$- \frac{8}{x^{3}} + \frac{x}{2}$

Step 5.3

Reorder terms.

$\frac{x}{2} - \frac{8}{x^{3}}$