Differentiation — Chapter Summary

The derivative of f(x) at x

The derivative of f(x) at x is defined by f'(x)=lim_{h\to0}(f(x+h)-f(x))/h. This is the direct limit definition and is also called differentiation fro

Important rules include d/dx(a)=0 for

Important rules include d/dx(a)=0 for a constant, d/dx(x^n)=nx^{n-1}, sum and difference rules, constant multiple rule, product rule, and quotient rul

For a composite function f(g(x))

For a composite function f(g(x)), the derivative is (f\circ g)'(x)=f'(g(x))g'(x). It is also written as dy/dx=(dy/du)(du/dx).

The derivatives are d/dx(e^x)=e^x

The derivatives are d/dx(e^x)=e^x, d/dx(a^x)=a^x log a, and d/dx(log_e x)=1/x. The proof of e^x uses first principle and the limit (e^h-1)/h=1.

Standard derivatives include d/dx(sin x)=cos x

Standard derivatives include d/dx(sin x)=cos x, d/dx(cos x)=-sin x, d/dx(tan x)=sec^2 x, d/dx(cot x)=-cosec^2 x, d/dx(sec x)=sec x tan x, and d/dx(cos