Techniques of Differentiation

 

 

 

Contents

 

Differentiation

  

Sum & Difference Rules

  

Product Rule

  

Quotient Rule

  

Chain Rule

  

Exponential Function

  

Logarithmic Function

  

Trigonometric Function

  

Reciprocal Function

  

Inverse Function

  

Implicit Differentiation

  

Higher Derivatives

  

Quiz

  

Calculus Main Page

Sum & Difference Rules

 
 
Now that you know how to differentiate a single term, what about a polynomial in many terms? This is where the sum and difference rules come in
 
Sum Rule:
    Given y = f(x) + g(x), then
                                             
 
The difference rule is a special case of the sum rule.
Difference Rule:
    Given y = f(x) - g(x), then
                                            
 
 
Here are some examples to illustrate the rules:
Differentiate the following w.r.t x:
           (i)   y = 3x2 + 2x + 1
           (ii)  y = x3 - 4x2 + 3
           (iii) y = x4 - x2
 
Solution:
    (i)   dy/dx = 6x + 2 = 2 (3x + 1)
    (ii)  dy/dx = 3x2 - 8x = x (3x - 8)
    (iii) dy/dx = 4x3 + 2x = 2x (2x2 + 1)
Factorization is not a necessary step, but it can simplify calculations especially if the differentiation is in a long and complicated question.