Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x^{2}}{3})$

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

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{2}]$ .

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

Step 3.2

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

$\frac{1}{3} (2 x)$

Step 3.3

Combine fractions.

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Step 3.3.1

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3} x$

Step 3.3.2

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

$\frac{2 x}{3}$

Step 4

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

$y' = \frac{2 x}{3}$

Step 5

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

$\frac{d y}{d x} = \frac{2 x}{3}$