Power Rule

\frac{d}{d x} x^{n} = n x^{n - 1}

Derivative of a power of x. Valid for any real n (x > 0 if n is not an integer). Linearity lets you differentiate polynomials term by term. The reverse is the antiderivative x^{n+1}/(n+1) for n ≠ −1.

$n$: Constant exponent
$x$: Independent variable
$d / d x$: Derivative operator

Step-by-step solved example

Evaluate d/dx (3x^4) at x = 2.

Slope of x^n is n x^{n−1} — a tangent on the graph, like dσ/dε = E.

1. Differentiate
d/dx (3x^4) = 12 x^3.
$12 x^{3}$
2. Evaluate
At x = 2: 12(8) = 96.
$96$

Answer: 96

10 practice questions

0/10 correct

1.d/dx (x^3) equals

2.d/dx (x^5) at x = 1 is

3.d/dx (4x) equals

4.d/dx (√x) equals

5.d/dx of a constant equals

6.d/dx (x^{−1}) equals

7.d/dx (x^4) at x = 2 is

8.d²/dx² (x^3) equals

9.Power rule with n = 0 gives d/dx (1) =

10.d/dx (2x^3) at x = 1 is