Differentiation Types

(i) Algebraic functions:

(ii) Exponential functions:

(iii) Logarithmic functions:

(iv) Trigonometric functions:

a) d/dx (sin x) = cosx

b) d/dx (cos x) = - sin x

c) d/dx (tan x) = sec²x

d) d/dx (cot x) = - cosec²x

e) d/dx (sec x) = sec x. tan x

f) d/dx (cosec x) = - cosec x. cot x

(v) Inverse Trigonometric functions:

Logarithmic differentiation: In this section, we will be mainly discussing derivatives of the function of the form [f(x)]^g(x) where f(x) and g(x) are functions of x To find the derivative of this type of function we proceed as follows:

Let y = [f(x)]^g(x)

Taking logarithm of both the sides, we have

logy = g(x). log [f(x)]

Differentiating w.r.t x we get

Alternative:

We have,

Example: Find the derivative of x

Solution: Let y = x^x

logy = xlogx

Differentiate w.r.t x

1/y dy/dx = logx + x.1/x

dy/dx = y [logx + 1]

∴ dy/dx = x^x [log x + 1]