Calculus Examples

$9 x^{3} - \frac{2}{3} x^{2} + \sqrt{x}$

Step 1

By the Sum Rule, the derivative of $9 x^{3} - \frac{2}{3} x^{2} + \sqrt{x}$ with respect to $x$ is $\frac{d}{d x} [9 x^{3}] + \frac{d}{d x} [- \frac{2}{3} x^{2}] + \frac{d}{d x} [\sqrt{x}]$ .

$\frac{d}{d x} [9 x^{3}] + \frac{d}{d x} [- \frac{2}{3} x^{2}] + \frac{d}{d x} [\sqrt{x}]$

Step 2

Evaluate $\frac{d}{d x} [9 x^{3}]$ .

Tap for more steps...

Step 2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{3}$ with respect to $x$ is $9 \frac{d}{d x} [x^{3}]$ .

$9 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{2}{3} x^{2}] + \frac{d}{d x} [\sqrt{x}]$

Step 2.2

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

$9 (3 x^{2}) + \frac{d}{d x} [- \frac{2}{3} x^{2}] + \frac{d}{d x} [\sqrt{x}]$

Step 2.3

Multiply $3$ by $9$ .

$27 x^{2} + \frac{d}{d x} [- \frac{2}{3} x^{2}] + \frac{d}{d x} [\sqrt{x}]$

Step 3

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

Tap for more steps...

Step 3.1

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

$27 x^{2} - \frac{2}{3} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\sqrt{x}]$

Step 3.2

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

$27 x^{2} - \frac{2}{3} (2 x) + \frac{d}{d x} [\sqrt{x}]$

Step 3.3

Multiply $2$ by $- 1$ .

$27 x^{2} - 2 (\frac{2}{3}) x + \frac{d}{d x} [\sqrt{x}]$

Step 3.4

Combine $- 2$ and $\frac{2}{3}$ .

$27 x^{2} + \frac{- 2 \cdot 2}{3} x + \frac{d}{d x} [\sqrt{x}]$

Step 3.5

Multiply $- 2$ by $2$ .

$27 x^{2} + \frac{- 4}{3} x + \frac{d}{d x} [\sqrt{x}]$

Step 3.6

Combine $\frac{- 4}{3}$ and $x$ .

$27 x^{2} + \frac{- 4 x}{3} + \frac{d}{d x} [\sqrt{x}]$

Step 3.7

Move the negative in front of the fraction.

$27 x^{2} - \frac{4 x}{3} + \frac{d}{d x} [\sqrt{x}]$

Step 4

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

Tap for more steps...

Step 4.1

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

$27 x^{2} - \frac{4 x}{3} + \frac{d}{d x} [x^{\frac{1}{2}}]$

Step 4.2

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

$27 x^{2} - \frac{4 x}{3} + \frac{1}{2} x^{\frac{1}{2} - 1}$

Step 4.3

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

$27 x^{2} - \frac{4 x}{3} + \frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 4.4

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

$27 x^{2} - \frac{4 x}{3} + \frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 4.5

Combine the numerators over the common denominator.

$27 x^{2} - \frac{4 x}{3} + \frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}$

Step 4.6

Simplify the numerator.

Tap for more steps...

Step 4.6.1

Multiply $- 1$ by $2$ .

$27 x^{2} - \frac{4 x}{3} + \frac{1}{2} x^{\frac{1 - 2}{2}}$

Step 4.6.2

Subtract $2$ from $1$ .

$27 x^{2} - \frac{4 x}{3} + \frac{1}{2} x^{\frac{- 1}{2}}$

Step 4.7

Move the negative in front of the fraction.

$27 x^{2} - \frac{4 x}{3} + \frac{1}{2} x^{- \frac{1}{2}}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

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

$27 x^{2} - \frac{4 x}{3} + \frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}}$

Step 5.2

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

$27 x^{2} - \frac{4 x}{3} + \frac{1}{2 x^{\frac{1}{2}}}$