Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Combine $4$ and $\frac{2}{3}$ .

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

Step 2.4

Multiply $4$ by $2$ .

$\frac{8}{3} x^{3} + \frac{d}{d x} [5 x] + \frac{d}{d x} [- x^{- 3}]$

Step 2.5

Combine $\frac{8}{3}$ and $x^{3}$ .

$\frac{8 x^{3}}{3} + \frac{d}{d x} [5 x] + \frac{d}{d x} [- x^{- 3}]$

Step 3

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

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Step 3.1

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

$\frac{8 x^{3}}{3} + 5 \frac{d}{d x} [x] + \frac{d}{d x} [- x^{- 3}]$

Step 3.2

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

$\frac{8 x^{3}}{3} + 5 \cdot 1 + \frac{d}{d x} [- x^{- 3}]$

Step 3.3

Multiply $5$ by $1$ .

$\frac{8 x^{3}}{3} + 5 + \frac{d}{d x} [- x^{- 3}]$

Step 4

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

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Step 4.1

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

$\frac{8 x^{3}}{3} + 5 - \frac{d}{d x} [x^{- 3}]$

Step 4.2

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

$\frac{8 x^{3}}{3} + 5 - (- 3 x^{- 4})$

Step 4.3

Multiply $- 3$ by $- 1$ .

$\frac{8 x^{3}}{3} + 5 + 3 x^{- 4}$

Step 5

Simplify.

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Step 5.1

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

$\frac{8 x^{3}}{3} + 5 + 3 \frac{1}{x^{4}}$

Step 5.2

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

$\frac{8 x^{3}}{3} + 5 + \frac{3}{x^{4}}$

Step 5.3

Reorder terms.

$\frac{8 x^{3}}{3} + \frac{3}{x^{4}} + 5$