Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Combine $3$ and $\frac{5}{3}$ .

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

Step 2.4

Multiply $3$ by $5$ .

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

Step 2.5

Combine $\frac{15}{3}$ and $x^{2}$ .

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

Step 2.6

Cancel the common factor of $15$ and $3$ .

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Step 2.6.1

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

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

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $3$ out of $3$ .

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

Step 2.6.2.2

Cancel the common factor.

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

Step 2.6.2.3

Rewrite the expression.

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

Step 2.6.2.4

Divide $5 x^{2}$ by $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

$5 x^{2} - 2 \cdot 1$

Step 3.3

Multiply $- 2$ by $1$ .

$5 x^{2} - 2$