Fundamental rules for differentiation

Rule (I): Differentiation of a constant function is zero i.e., d/dx (c) = 0

Rule (II): Let f(x) be a differentiable function and let c be a constant. Then c.f(x) is also differentiable such that 

This is the derivative of a constant times a function is the constant times the derivative of the function.

Rule (III): If f(x) and g(x) are differentiable functions, then show that f(x) ± g(x) are also differentiable such that 

That is the derivative of the sum or difference of two functions is the sum or difference of their derivatives.

Rule (IV): If f(x) and g(x) are two differentiable functions, then f(x).g(x) is also differentiable such that 

Rule (V) (Quotient Rule): If f(x) and g(x) are two differentiable functions and g(x) ≠ 0 then f(x)/g(x) is also differentiable such that 

Relation between dy/dx and dx/dy:

Let x and y be two variables connected by a relation of the form f(x, y) = 0. Let Δx be a small change in x and let Δy be the corresponding change in y Then  and 

Now,

Ex: Differentiate f(x) = sinx.logx

Solution: Using product rule of differentiation

= logx.cosx + sinx/x

Ex: Find the derivative of f(x) = logx/x²

Solution: Using quotient rule of differentiation