Calculus Examples

$2.8 x^{- 3} - \frac{0.6}{3 \sqrt{x^{2}}} + 7$

Step 1

By the Sum Rule, the derivative of $2.8 x^{- 3} - \frac{0.6}{3 \sqrt{x^{2}}} + 7$ with respect to $x$ is $\frac{d}{d x} [2.8 x^{- 3}] + \frac{d}{d x} [- \frac{0.6}{3 \sqrt{x^{2}}}] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [2.8 x^{- 3}] + \frac{d}{d x} [- \frac{0.6}{3 \sqrt{x^{2}}}] + \frac{d}{d x} [7]$

Step 2

Evaluate $\frac{d}{d x} [2.8 x^{- 3}]$ .

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Step 2.1

Since $2.8$ is constant with respect to $x$ , the derivative of $2.8 x^{- 3}$ with respect to $x$ is $2.8 \frac{d}{d x} [x^{- 3}]$ .

$2.8 \frac{d}{d x} [x^{- 3}] + \frac{d}{d x} [- \frac{0.6}{3 \sqrt{x^{2}}}] + \frac{d}{d x} [7]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$2.8 (- 3 x^{- 4}) + \frac{d}{d x} [- \frac{0.6}{3 \sqrt{x^{2}}}] + \frac{d}{d x} [7]$

Step 2.3

Multiply $- 3$ by $2.8$ .

$- 8.4 x^{- 4} + \frac{d}{d x} [- \frac{0.6}{3 \sqrt{x^{2}}}] + \frac{d}{d x} [7]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{0.6}{3 \sqrt{x^{2}}}]$ .

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Step 3.1

Pull terms out from under the radical, assuming positive real numbers.

$- 8.4 x^{- 4} + \frac{d}{d x} [- \frac{0.6}{3 x}] + \frac{d}{d x} [7]$

Step 3.2

Since $- \frac{0.6}{3}$ is constant with respect to $x$ , the derivative of $- \frac{0.6}{3 x}$ with respect to $x$ is $- \frac{0.6}{3} \frac{d}{d x} [\frac{1}{x}]$ .

$- 8.4 x^{- 4} - \frac{0.6}{3} \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [7]$

Step 3.3

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

$- 8.4 x^{- 4} - \frac{0.6}{3} \frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [7]$

Step 3.4

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

$- 8.4 x^{- 4} - \frac{0.6}{3} (- x^{- 2}) + \frac{d}{d x} [7]$

Step 3.5

Multiply $- 1$ by $- 1$ .

$- 8.4 x^{- 4} + 1 (\frac{0.6}{3}) x^{- 2} + \frac{d}{d x} [7]$

Step 3.6

Multiply $\frac{0.6}{3}$ by $1$ .

$- 8.4 x^{- 4} + \frac{0.6}{3} x^{- 2} + \frac{d}{d x} [7]$

Step 3.7

Combine $\frac{0.6}{3}$ and $x^{- 2}$ .

$- 8.4 x^{- 4} + \frac{0.6 x^{- 2}}{3} + \frac{d}{d x} [7]$

Step 3.8

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 8.4 x^{- 4} + \frac{0.6}{3 x^{2}} + \frac{d}{d x} [7]$

Step 4

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$- 8.4 x^{- 4} + \frac{0.6}{3 x^{2}} + 0$

Step 5

Simplify.

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Step 5.1

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

$- 8.4 \frac{1}{x^{4}} + \frac{0.6}{3 x^{2}} + 0$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $- 8.4$ and $\frac{1}{x^{4}}$ .

$\frac{- 8.4}{x^{4}} + \frac{0.6}{3 x^{2}} + 0$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{8.4}{x^{4}} + \frac{0.6}{3 x^{2}} + 0$

Step 5.2.3

Add $- \frac{8.4}{x^{4}} + \frac{0.6}{3 x^{2}}$ and $0$ .

$- \frac{8.4}{x^{4}} + \frac{0.6}{3 x^{2}}$

Step 5.3

Reorder terms.

$\frac{0.6}{3 x^{2}} - \frac{8.4}{x^{4}}$

Step 5.4

Simplify each term.

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Step 5.4.1

Factor $0.6$ out of $0.6$ .

$\frac{0.6 (1)}{3 x^{2}} - \frac{8.4}{x^{4}}$

Step 5.4.2

Factor $3$ out of $3 x^{2}$ .

$\frac{0.6 (1)}{3 (x^{2})} - \frac{8.4}{x^{4}}$

Step 5.4.3

Separate fractions.

$\frac{0.6}{3} \cdot \frac{1}{x^{2}} - \frac{8.4}{x^{4}}$

Step 5.4.4

Divide $0.6$ by $3$ .

$0.2 \frac{1}{x^{2}} - \frac{8.4}{x^{4}}$

Step 5.4.5

Combine $0.2$ and $\frac{1}{x^{2}}$ .

$\frac{0.2}{x^{2}} - \frac{8.4}{x^{4}}$