Calculus Examples

$f (x) = \frac{x^{3}}{3}$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Simplify terms.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Cancel the common factor of $3$ .

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Step 1.3.3.1

Cancel the common factor.

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

Step 1.3.3.2

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

$f' (x) = x^{2}$

Step 2

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

$f'' (x) = 2 x$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $2 x$ .

$2 x$