Calculus Examples

$f (x) = 2 x^{\frac{4}{3}} - \frac{2}{3} x^{\frac{3}{2}} + x^{2} - 4 x + 1$

Step 1

By the Sum Rule, the derivative of $2 x^{\frac{4}{3}} - \frac{2}{3} x^{\frac{3}{2}} + x^{2} - 4 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{\frac{4}{3}}] + \frac{d}{d x} [- \frac{2}{3} x^{\frac{3}{2}}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$ .

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

Step 2

Evaluate $\frac{d}{d x} [2 x^{\frac{4}{3}}]$ .

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{\frac{4}{3}}$ with respect to $x$ is $2 \frac{d}{d x} [x^{\frac{4}{3}}]$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

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

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.4

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $3$ .

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

Step 2.6.2

Subtract $3$ from $4$ .

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

Step 2.7

Combine $\frac{4}{3}$ and $x^{\frac{1}{3}}$ .

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

Step 2.8

Combine $2$ and $\frac{4 x^{\frac{1}{3}}}{3}$ .

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

Step 2.9

Multiply $4$ by $2$ .

$\frac{8 x^{\frac{1}{3}}}{3} + \frac{d}{d x} [- \frac{2}{3} x^{\frac{3}{2}}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3

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

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Step 3.1

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

$\frac{8 x^{\frac{1}{3}}}{3} - \frac{2}{3} \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{8 x^{\frac{1}{3}}}{3} - \frac{2}{3} (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3.3

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

$\frac{8 x^{\frac{1}{3}}}{3} - \frac{2}{3} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3.4

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

$\frac{8 x^{\frac{1}{3}}}{3} - \frac{2}{3} (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3.5

Combine the numerators over the common denominator.

$\frac{8 x^{\frac{1}{3}}}{3} - \frac{2}{3} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

$\frac{8 x^{\frac{1}{3}}}{3} - \frac{2}{3} (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3.6.2

Subtract $2$ from $3$ .

$\frac{8 x^{\frac{1}{3}}}{3} - \frac{2}{3} (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3.7

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

$\frac{8 x^{\frac{1}{3}}}{3} - \frac{2}{3} \cdot \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3.8

Multiply $\frac{3 x^{\frac{1}{2}}}{2}$ by $\frac{2}{3}$ .

$\frac{8 x^{\frac{1}{3}}}{3} - \frac{3 x^{\frac{1}{2}} \cdot 2}{2 \cdot 3} + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3.9

Multiply $2$ by $3$ .

$\frac{8 x^{\frac{1}{3}}}{3} - \frac{6 x^{\frac{1}{2}}}{2 \cdot 3} + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3.10

Multiply $2$ by $3$ .

$\frac{8 x^{\frac{1}{3}}}{3} - \frac{6 x^{\frac{1}{2}}}{6} + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3.11

Cancel the common factor.

$\frac{8 x^{\frac{1}{3}}}{3} - \frac{6 x^{\frac{1}{2}}}{6} + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 3.12

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

$\frac{8 x^{\frac{1}{3}}}{3} - x^{\frac{1}{2}} + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 4

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

$\frac{8 x^{\frac{1}{3}}}{3} - x^{\frac{1}{2}} + 2 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 5

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

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Step 5.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$\frac{8 x^{\frac{1}{3}}}{3} - x^{\frac{1}{2}} + 2 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 5.2

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

$\frac{8 x^{\frac{1}{3}}}{3} - x^{\frac{1}{2}} + 2 x - 4 \cdot 1 + \frac{d}{d x} [1]$

Step 5.3

Multiply $- 4$ by $1$ .

$\frac{8 x^{\frac{1}{3}}}{3} - x^{\frac{1}{2}} + 2 x - 4 + \frac{d}{d x} [1]$

Step 6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{8 x^{\frac{1}{3}}}{3} - x^{\frac{1}{2}} + 2 x - 4 + 0$

Step 7

Simplify.

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Step 7.1

Add $\frac{8 x^{\frac{1}{3}}}{3} - x^{\frac{1}{2}} + 2 x - 4$ and $0$ .

$\frac{8 x^{\frac{1}{3}}}{3} - x^{\frac{1}{2}} + 2 x - 4$

Step 7.2

Reorder terms.

$2 x - x^{\frac{1}{2}} + \frac{8 x^{\frac{1}{3}}}{3} - 4$