1. Basic Derivative formulae
(xn)0=nxn−1
(ax)0=axln a(ex)0=ex
(logax)0=1
xln a(ln x)0=1
x
(sin x)0= cos x(cos x)0=−sin x
(tan x)0= sec2x(cot x)0=−csc2x
(sec x)0= sec xtan x(csc x)0=−csc xcot x
(sin−1x)0=1
√1−x2(cos−1x)0=−1
√1−x2
(tan−1x)0=1
1 + x2(cot−1x)0=−1
1 + x2
(sec−1x)0=1
x√x2−1(csc−1x)0=−1
x√x2−1
2. Differentiation Rules
Sum rule: (f+g)0=f0+g0wheref=f(x), g =g(x)
Product rule: (f·g)0=f0g+g0f
Quotient rule: f
g0
=f0g−g0f
g2
Chain rule: [f(g)]0=f0(g)·g0or dy
dx =dy
du ·du
dx
Implicit differentiation:
If y=y(x) is given implicitly, find derivative to the entire equation with respect to x. Then solve for y0.
3. Identities of Trigonometric Functions
tan x=sin x
cos xcot x=cos x
sin x
sec x=1
cos xcsc x=1
sin x
sin2x+ cos2x= 1 1 + tan2x= sec2x1 + cot2x= csc2x
4. Laws of Exponential Functions and Logarithms Functions
ax·ay=ex+yloga(xy) = loga(x) + loga(y)
ax
ay=ax−yloga(x
y) = loga(x)−loga(y)
(ax)y=axy loga(xn) = nloga(x)
aloga(x)=xln x= logex
a0= 1 logaa= 1,loga1 = 0
1
