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

Higher Derivatives

 
 
Higher Derivatives
Higher derivatives are obtained by repeatedly differentiating an expression or equation.
 
First derivative:
                    
Second derivative:
                    
This is obtained by differentiating y with respect to x twice.
Third derivative:
                    
This is obtained by differentiating y with respect to x thrice.
 
Obtaining higher derivatives is relative simple. Consider the equation
                  y = x5 + 3x4 + 2x3 -4x - 7
                       
 
Implicit Functions
For implicit functions, the process is similar, just differentiate the equation repeatedly.
 
                x3 + 4y2 + 3xy2 = y + 7
First derivative:
                       
Second derivative:
               
 
Applications
The idea of higher derivatives is very useful in calculus, some of which we explain in our website. It is used in kinematics, rates of change as well as curve sketching and determination of stationary points. Other applications include differential equations and power series.