Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $1$ .

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

Step 3

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

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3.2

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

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

Step 3.3

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 3.3.1

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

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

Step 3.3.2

Multiply $x^{- \frac{1}{2}}$ by $x$ .

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Step 3.3.2.1

Raise $x$ to the power of $1$ .

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

Step 3.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.3

Write $1$ as a fraction with a common denominator.

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

Step 3.3.4

Combine the numerators over the common denominator.

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

Step 3.3.5

Add $- 1$ and $2$ .

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

Step 3.4

Multiply $3$ by $- 1$ .

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

Step 3.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{\frac{1}{2}}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Combine the numerators over the common denominator.

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

Step 3.10

Simplify the numerator.

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Step 3.10.1

Multiply $- 1$ by $2$ .

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

Step 3.10.2

Subtract $2$ from $1$ .

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

Step 3.11

Move the negative in front of the fraction.

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

Step 3.12

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

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

Step 3.13

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

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

Step 3.14

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

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

Step 3.15

Move the negative in front of the fraction.

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