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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} (x^n) = nx^{n-1} }

for all integers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } . This rule is useful when combined with the chain rule. As an example we can compute the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x) = \lim_{h\rarr0} \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x) = nx^{n-1}. }

QED