Calculus Examples

$f (x) = 20 x^{- 0.6} + 32^{- 4}$

Step 1

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

$\frac{d}{d x} [20 x^{- 0.6} + \frac{1}{32^{4}}]$

Step 2

Differentiate using the Sum Rule.

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Step 2.1

Raise $32$ to the power of $4$ .

$\frac{d}{d x} [20 x^{- 0.6} + \frac{1}{1048576}]$

Step 2.2

By the Sum Rule, the derivative of $20 x^{- 0.6} + \frac{1}{1048576}$ with respect to $x$ is $\frac{d}{d x} [20 x^{- 0.6}] + \frac{d}{d x} [\frac{1}{1048576}]$ .

$\frac{d}{d x} [20 x^{- 0.6}] + \frac{d}{d x} [\frac{1}{1048576}]$

Step 3

Evaluate $\frac{d}{d x} [20 x^{- 0.6}]$ .

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Step 3.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x^{- 0.6}$ with respect to $x$ is $20 \frac{d}{d x} [x^{- 0.6}]$ .

$20 \frac{d}{d x} [x^{- 0.6}] + \frac{d}{d x} [\frac{1}{1048576}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 0.6$ .

$20 (- 0.6 x^{- 1.6}) + \frac{d}{d x} [\frac{1}{1048576}]$

Step 3.3

Multiply $- 0.6$ by $20$ .

$- 12 x^{- 1.6} + \frac{d}{d x} [\frac{1}{1048576}]$

Step 4

Since $\frac{1}{1048576}$ is constant with respect to $x$ , the derivative of $\frac{1}{1048576}$ with respect to $x$ is $0$ .

$- 12 x^{- 1.6} + 0$

Step 5

Simplify.

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Step 5.1

Add $- 12 x^{- 1.6}$ and $0$ .

$- 12 x^{- 1.6}$

Step 5.2

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

$- 12 \frac{1}{x^{1.6}}$

Step 5.3

Combine $- 12$ and $\frac{1}{x^{1.6}}$ .

$\frac{- 12}{x^{1.6}}$

Step 5.4

Move the negative in front of the fraction.

$- \frac{12}{x^{1.6}}$