Differentiation — Formula Sheet

f'(x)=lim_{h→0}((f(x+h)-f(x))/h)

Df(x) or d/dx[f(x)]

d/dx(f+g)=df/dx+dg/dx

d/dx(f-g)=df/dx-dg/dx

d/dx(cf)=c(df/dx)

d/dx(F1F2)=F1(dF2/dx)+F2(dF1/dx)

d/dx(F2/F1)=(F1(dF2/dx)-F2(dF1/dx))/(F1^2)

d/dx(e^x)=e^x

d/dx(a^x)=a^x log a

d/dx(log_e x)=1/x

d/dx(e^F)=e^F(dF/dx)

d/dx(N^F)=N^F(log N)(dF/dx)

(f∘g)'(x)=f'(g(x))g'(x)

dy/dx=(dy/du)·(du/dx)

An implicit function is defined by a relation f(x,y)=0 rather than by y alone.

d/dx(e^x)=e^x.

The derivative of f(x) at x is defined by f'(x)=lim_{h\to0}(f(x+h)-f(x))/h.

It is also written as dy/dx=(dy/du)(du/dx).

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.