Calculus Examples

$y = - \frac{1}{2} x^{4} + 3 x^{\frac{5}{3}} + 2 x$

Step 1

By the Sum Rule, the derivative of $- \frac{1}{2} x^{4} + 3 x^{\frac{5}{3}} + 2 x$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{2} x^{4}] + \frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$ .

$\frac{d}{d x} [- \frac{1}{2} x^{4}] + \frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$

Step 2

Evaluate $\frac{d}{d x} [- \frac{1}{2} x^{4}]$ .

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Step 2.1

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{1}{2} x^{4}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [x^{4}]$ .

$- \frac{1}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$

Step 2.2

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

$- \frac{1}{2} (4 x^{3}) + \frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$

Step 2.3

Multiply $4$ by $- 1$ .

$- 4 (\frac{1}{2}) x^{3} + \frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$

Step 2.4

Combine $- 4$ and $\frac{1}{2}$ .

$\frac{- 4}{2} x^{3} + \frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$

Step 2.5

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

$\frac{- 4 x^{3}}{2} + \frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$

Step 2.6

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

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Step 2.6.1

Factor $2$ out of $- 4 x^{3}$ .

$\frac{2 (- 2 x^{3})}{2} + \frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $2$ out of $2$ .

$\frac{2 (- 2 x^{3})}{2 (1)} + \frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$

Step 2.6.2.2

Cancel the common factor.

$\frac{2 (- 2 x^{3})}{2 \cdot 1} + \frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$

Step 2.6.2.3

Rewrite the expression.

$\frac{- 2 x^{3}}{1} + \frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$

Step 2.6.2.4

Divide $- 2 x^{3}$ by $1$ .

$- 2 x^{3} + \frac{d}{d x} [3 x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$

Step 3

Evaluate $\frac{d}{d x} [3 x^{\frac{5}{3}}]$ .

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Step 3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{\frac{5}{3}}$ with respect to $x$ is $3 \frac{d}{d x} [x^{\frac{5}{3}}]$ .

$- 2 x^{3} + 3 \frac{d}{d x} [x^{\frac{5}{3}}] + \frac{d}{d x} [2 x]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{3}$ .

$- 2 x^{3} + 3 (\frac{5}{3} x^{\frac{5}{3} - 1}) + \frac{d}{d x} [2 x]$

Step 3.3

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

$- 2 x^{3} + 3 (\frac{5}{3} x^{\frac{5}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [2 x]$

Step 3.4

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

$- 2 x^{3} + 3 (\frac{5}{3} x^{\frac{5}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [2 x]$

Step 3.5

Combine the numerators over the common denominator.

$- 2 x^{3} + 3 (\frac{5}{3} x^{\frac{5 - 1 \cdot 3}{3}}) + \frac{d}{d x} [2 x]$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $3$ .

$- 2 x^{3} + 3 (\frac{5}{3} x^{\frac{5 - 3}{3}}) + \frac{d}{d x} [2 x]$

Step 3.6.2

Subtract $3$ from $5$ .

$- 2 x^{3} + 3 (\frac{5}{3} x^{\frac{2}{3}}) + \frac{d}{d x} [2 x]$

Step 3.7

Combine $\frac{5}{3}$ and $x^{\frac{2}{3}}$ .

$- 2 x^{3} + 3 \frac{5 x^{\frac{2}{3}}}{3} + \frac{d}{d x} [2 x]$

Step 3.8

Combine $3$ and $\frac{5 x^{\frac{2}{3}}}{3}$ .

$- 2 x^{3} + \frac{3 (5 x^{\frac{2}{3}})}{3} + \frac{d}{d x} [2 x]$

Step 3.9

Multiply $5$ by $3$ .

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

Step 3.10

Factor $3$ out of $15 x^{\frac{2}{3}}$ .

$- 2 x^{3} + \frac{3 (5 x^{\frac{2}{3}})}{3} + \frac{d}{d x} [2 x]$

Step 3.11

Cancel the common factors.

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Step 3.11.1

Factor $3$ out of $3$ .

$- 2 x^{3} + \frac{3 (5 x^{\frac{2}{3}})}{3 (1)} + \frac{d}{d x} [2 x]$

Step 3.11.2

Cancel the common factor.

$- 2 x^{3} + \frac{3 (5 x^{\frac{2}{3}})}{3 \cdot 1} + \frac{d}{d x} [2 x]$

Step 3.11.3

Rewrite the expression.

$- 2 x^{3} + \frac{5 x^{\frac{2}{3}}}{1} + \frac{d}{d x} [2 x]$

Step 3.11.4

Divide $5 x^{\frac{2}{3}}$ by $1$ .

$- 2 x^{3} + 5 x^{\frac{2}{3}} + \frac{d}{d x} [2 x]$

Step 4

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

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Step 4.1

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

$- 2 x^{3} + 5 x^{\frac{2}{3}} + 2 \frac{d}{d x} [x]$

Step 4.2

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

$- 2 x^{3} + 5 x^{\frac{2}{3}} + 2 \cdot 1$

Step 4.3

Multiply $2$ by $1$ .

$- 2 x^{3} + 5 x^{\frac{2}{3}} + 2$