Calculus Examples

$y = 30 x^{- \frac{2}{4}} + 3 x$

Step 1

By the Sum Rule, the derivative of $30 x^{- \frac{2}{4}} + 3 x$ with respect to $x$ is $\frac{d}{d x} [30 x^{- \frac{2}{4}}] + \frac{d}{d x} [3 x]$ .

$\frac{d}{d x} [30 x^{- \frac{2}{4}}] + \frac{d}{d x} [3 x]$

Step 2

Evaluate $\frac{d}{d x} [30 x^{- \frac{2}{4}}]$ .

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Step 2.1

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

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Step 2.1.1

Factor $2$ out of $2$ .

$\frac{d}{d x} [30 x^{- \frac{2 (1)}{4}}] + \frac{d}{d x} [3 x]$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $2$ out of $4$ .

$\frac{d}{d x} [30 x^{- \frac{2 \cdot 1}{2 \cdot 2}}] + \frac{d}{d x} [3 x]$

Step 2.1.2.2

Cancel the common factor.

$\frac{d}{d x} [30 x^{- \frac{2 \cdot 1}{2 \cdot 2}}] + \frac{d}{d x} [3 x]$

Step 2.1.2.3

Rewrite the expression.

$\frac{d}{d x} [30 x^{- \frac{1}{2}}] + \frac{d}{d x} [3 x]$

Step 2.2

Since $30$ is constant with respect to $x$ , the derivative of $30 x^{- \frac{1}{2}}$ with respect to $x$ is $30 \frac{d}{d x} [x^{- \frac{1}{2}}]$ .

$30 \frac{d}{d x} [x^{- \frac{1}{2}}] + \frac{d}{d x} [3 x]$

Step 2.3

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

$30 (- \frac{1}{2} x^{- \frac{1}{2} - 1}) + \frac{d}{d x} [3 x]$

Step 2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$30 (- \frac{1}{2} x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [3 x]$

Step 2.5

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

$30 (- \frac{1}{2} x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [3 x]$

Step 2.6

Combine the numerators over the common denominator.

$30 (- \frac{1}{2} x^{\frac{- 1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [3 x]$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

$30 (- \frac{1}{2} x^{\frac{- 1 - 2}{2}}) + \frac{d}{d x} [3 x]$

Step 2.7.2

Subtract $2$ from $- 1$ .

$30 (- \frac{1}{2} x^{\frac{- 3}{2}}) + \frac{d}{d x} [3 x]$

Step 2.8

Move the negative in front of the fraction.

$30 (- \frac{1}{2} x^{- \frac{3}{2}}) + \frac{d}{d x} [3 x]$

Step 2.9

Combine $x^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$30 (- \frac{x^{- \frac{3}{2}}}{2}) + \frac{d}{d x} [3 x]$

Step 2.10

Multiply $- 1$ by $30$ .

$- 30 \frac{x^{- \frac{3}{2}}}{2} + \frac{d}{d x} [3 x]$

Step 2.11

Combine $- 30$ and $\frac{x^{- \frac{3}{2}}}{2}$ .

$\frac{- 30 x^{- \frac{3}{2}}}{2} + \frac{d}{d x} [3 x]$

Step 2.12

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 30}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [3 x]$

Step 2.13

Factor $2$ out of $- 30$ .

$\frac{2 \cdot - 15}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [3 x]$

Step 2.14

Cancel the common factors.

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Step 2.14.1

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

$\frac{2 \cdot - 15}{2 (x^{\frac{3}{2}})} + \frac{d}{d x} [3 x]$

Step 2.14.2

Cancel the common factor.

$\frac{2 \cdot - 15}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [3 x]$

Step 2.14.3

Rewrite the expression.

$\frac{- 15}{x^{\frac{3}{2}}} + \frac{d}{d x} [3 x]$

Step 2.15

Move the negative in front of the fraction.

$- \frac{15}{x^{\frac{3}{2}}} + \frac{d}{d x} [3 x]$

Step 3

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

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Step 3.1

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

$- \frac{15}{x^{\frac{3}{2}}} + 3 \frac{d}{d x} [x]$

Step 3.2

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

$- \frac{15}{x^{\frac{3}{2}}} + 3 \cdot 1$

Step 3.3

Multiply $3$ by $1$ .

$- \frac{15}{x^{\frac{3}{2}}} + 3$

Step 4

Reorder terms.

$3 - \frac{15}{x^{\frac{3}{2}}}$