Calculus Examples

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

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [\frac{1}{3} x^{3}]$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Cancel the common factor of $3$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

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

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

Step 3

Evaluate $\frac{d}{d x} [\frac{3}{5} x^{2}]$ .

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Step 3.1

Since $\frac{3}{5}$ is constant with respect to $x$ , the derivative of $\frac{3}{5} x^{2}$ with respect to $x$ is $\frac{3}{5} \frac{d}{d x} [x^{2}]$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $3$ .

$x^{2} + \frac{6}{5} x + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]$

Step 3.5

Combine $\frac{6}{5}$ and $x$ .

$x^{2} + \frac{6 x}{5} + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 3]$

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

$x^{2} + \frac{6 x}{5} + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 4.2

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

$x^{2} + \frac{6 x}{5} + 2 \cdot 1 + \frac{d}{d x} [- 3]$

Step 4.3

Multiply $2$ by $1$ .

$x^{2} + \frac{6 x}{5} + 2 + \frac{d}{d x} [- 3]$

Step 5

Differentiate using the Constant Rule.

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Step 5.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$x^{2} + \frac{6 x}{5} + 2 + 0$

Step 5.2

Add $x^{2} + \frac{6 x}{5} + 2$ and $0$ .

$x^{2} + \frac{6 x}{5} + 2$