What is the power rule for taking derivatives of polynomial functions?

[Greg Bernhardt]

Definition/Summary

A method used to take the derivative of a polynomial function.

Equations

$$\frac{d}{dx} x^{n} = nx^{n-1}$$

Extended explanation

Power rule applies to a function of the form $x^{n}$, where x is the variable and n is a constant. Used in combination with the sum and constant factor rules of differentiation, power rule can be a powerful tool for taking derivatives.

Proof:

We can apply the limit definition of a derivative to this specific function:
$$f'(x) := \lim_{h→0} \frac{f(x+h)-f(x)}{h}$$
Substituting in gives us:
$$\frac{d}{dx} x^{n} = \lim_{h→0} \frac{(x+h)^{n}-x^{n}}{h}$$
If we then expand using Binomial Theorem:
$$\frac{d}{dx} x^{n} = \lim_{h→0} \frac{x^{n}+nx^{n-1}h+\binom{n}{2}x^{n-2}h^{2}+\cdots+h^{n} -x^{n}}{h}$$
We can then cancel the first and last $x^{n}$ terms and distribute the h from the denominator:
$$\frac{d}{dx} x^{n} = \lim_{h→0} nx^{n-1}+\binom{n}{2}x^{n-2}h+\cdots+h^{n-1}$$
Finally, we take the limit by substituting in h=0:
$$\frac{d}{dx} x^{n} = nx^{n-1}+\binom{n}{2}x^{n-2}0+\cdots+0^{n-1}$$
$$\frac{d}{dx} x^{n} = nx^{n-1}$$

Example 1:
$$f(x) = x^{189}$$
$$f'(x) = 189x^{189-1} = 189x^{188}$$
Example 2:
$$f(x) = 3x^{3}+7x^{2}+8x+2$$
$$f'(x) = 9x^{2}+14x+8$$
Example 3:
$$f(x) = 3\sqrt{x}$$
$$f'(x) = 3×1/2\ x^{(1/2-1)} = \frac{3}{2\sqrt{x}}$$

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[fresh_42]

The power rule is a consecutive application of the product or Leibniz rule:
$$
\dfrac{d}{dx}x^n = \dfrac{d}{dx}(x\cdot x^{n-1})=\left( \dfrac{d}{dx} x\right) \cdot x^{n-1} + x\cdot \dfrac{d}{dx}x^{n-1}=1\cdot x^{n-1} + x\cdot (n-1)\cdot x^{n-2}=n \cdot x^{n-1}
$$
by induction.