List of derivatives

Posted February 9th, 2009 by Structure

We have

$f'(x)=\lim_{y\to x}\frac{f(y)-f(x)}{y-x}$

For many elementary functions this type of limit is very easy to be found. We need some algebra but sometimes you have to use some remarkable limits.

$f(x)=c\:f'(x)=0\; as \:\lim_{y\to x}\frac{c-c}{y-x}=0$

$f(x)=x\: f'(x)=1\:as \lim_{y\to x}\frac{y-x}{y-x}=1$

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

$\lim_{y\to x}\frac{y^n-x^n}{y-x}=\lim_{y\to x}y^{n-1}+y^{n-2}x+...+yx^{n-2}+x^{n-1}=nx^{n-1}$

$(lnx)'=\frac{1}{x}$

$(\sqrt x)'=\frac{1}{2\sqrt x};\lim_{y\to x}\frac{\sqrt y-\sqrt x}{y-x}=\lim_{y\to x}\frac{1}{\sqrt y+\sqrt x}=\frac{1}{2\sqrt x}$

$(\sqrt[3] x)'=\frac{1}{3\sqrt[3] {x^2}};\lim_{y\to x}\frac{\sqrt[3] y-\sqrt[3] x}{y-x}=\lim_{y\to x}\frac{1}{\sqrt [3]{y^2}+\sqrt[3 ]{yx}+\sqrt [3]{x^2}}=\frac{1}{3\sqrt[3]{ x^2}}$

$(\sqrt[n] x)'=\frac{1}{n\sqrt[n] {x^{n-1}}}$

$(e^x)'=e^x$

$(a^x)'=(e^{x\ln a})'=\ln ae^{x\ln a}=a^x ln a$

$(e^{f(x)})'=e^{f(x)}f'(x)$

$(\sin x)'=\cos x; (\sin ax)'=a\cos ax$

Proof

$(\sin x)'=\lim_{y\to x}\frac{\sin y-\sin x}{y-x}=\lim_{y\to x}\frac{2\sin\frac{y-x}{2}\cos \frac{y+x}{2}}{y-x}=\lim_{y\to x}\frac{\sin\frac{y-x}{2}}{\frac{y-x}{2}}\cos\frac{y+x}{2}=\cos x$

$(\cos x)'=-\sin x;(\cos ax)'=-a\sin ax$

Proof

$(\cos x)'=\lim_{y\to x}\frac{\cos y-\cos x}{y-x}=\lim_{y\to x}-\frac{2\sin\frac{y-x}{2}\sin \frac{y+x}{2}}{y-x}=-\lim_{y\to x}\frac{\sin\frac{y-x}{2}}{\frac{y-x}{2}}\sin\frac{y+x}{2}=-\sin x$