Calculus Examples

$x^{n}$

Step 1

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

$f' (x) = n x^{n - 1}$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

$n ((n - 1) x^{n - 1 - 1})$

Step 2.3

Simplify the expression.

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Step 2.3.1

Subtract $1$ from $- 1$ .

$n ((n - 1) x^{n - 2})$

Step 2.3.2

Reorder the factors of $n ((n - 1) x^{n - 2})$ .

$f'' (x) = x^{n - 2} n (n - 1)$