Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x^{3}}{3} - x)$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Cancel the common factor of $3$ .

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Step 3.2.5.1

Cancel the common factor.

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

Step 3.2.5.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Multiply $- 1$ by $1$ .

$x^{2} - 1$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = x^{2} - 1$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = x^{2} - 1$