Power rule

The power rule allows one to calculate the derivative of a power of a function in terms of the derivative of the function itself. The power rule states that

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

for all integers $n$. This rule is useful when combined with the chain rule. As an example we can compute the derivative of $f(x)=(\sin(x))^{n}$ as

$f'(x)={\frac {d}{dx}}(\sin(x))^{n}=n(\sin(x))^{n-1}\cos(x)$

Proof

The power rule is simple and elegant to prove with the definition of a derivative:

$f'(x)=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}.$

Substituting $f(x)=x^{n}$ gives

$f'(x)=\lim _{h\rightarrow 0}{\frac {(x+h)^{n}-x^{n}}{h}}.$

The two polynomials in the numerator can be factored out. It is left as a series, since n can be any integer.

$f'(x)=\lim _{h\rightarrow 0}{\frac {((x+h)-x)((x+h)^{n-1}+(x+h)^{n-2}x+(x+h)^{n-3}x^{2}+\dots +(x+h)x^{n-2}+x^{n-1})}{h}}.$

Then if the first factor is eliminated:

$f'(x)=\lim _{h\rightarrow 0}{\frac {(h)((x+h)^{n-1}+(x+h)^{n-2}x+(x+h)^{n-3}x^{2}\dots +(x+h)x^{n-2}+x^{n-1})}{h}}.$

Now, the "h" can be eliminated. This is important, since the denominator cannot go to zero':

$f'(x)=\lim _{h\rightarrow 0}(x+h)^{n-1}+(x+h)^{n-2}x+(x+h)^{n-3}x^{2}\dots +(x+h)x^{n-2}+x^{n-1}.$

With no "h" in the denominator, the limit can be evaluated by letting h=0.

$f'(x)=(x+0)^{n-1}+(x+0)^{n-2}x+(x+0)^{n-3}x^{2}\dots +(x+0)x^{n-2}+x^{n-1}.$:

$f'(x)=x^{n-1}+x^{n-1}+x^{n-1}\dots +x^{n-1}+x^{n-1}.$

We see that there are n terms, so:

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

QED

Rules for finding derivatives

• Power rule
• Constant-multiple rule
• sum rule
• Chain rule
• Product rule
• Quotient rule