All the results that we explain in this section can be proven using the first principles, which has been explained in the previous section.

 

The most important rule in differentiation is:

Given y = xⁿ where n is a real number,

                                                

 

This rule forms the base for differentiating all functions. Here are some examples to illustrate the rule:

 

Example:

Differentiate, with respect to x, the following functions:

      (i)  x3

      (ii) 3x⁴

 

Solution:

 (i)  d/dx (x3) = 3x^3-1 = 3x²

 (ii) d/dx (3x⁴) = 4.3x^4-1 = 12x³

 

 

Some special cases of this rule:

 

Given y = mx , where m is a constant,

                                               

 

Given y = c, where c is constant,

                                              

 

Also,

Given y = (ax + b)ⁿ,

                                             

In general, for y = uⁿ, where u is a function of x,

                                             

 

Example:

Differentiate, with respect to x,

   (i)  (3x + 1)²

   (ii) (4x² + 3x + 1)³

 

Solution:

 (i)  d/dx [(3x + 1)²] = 2. (3x + 1) .3 = 6 (3x + 1)

 (ii) d/dx [(4x² + 3x + 1)³] = 3 (4x² + 3x + 1) (8x + 3)              sum rule

 

 

Now, armed with the basic knowledge of differentiation, go forth and explore more.