Coupon Accepted Successfully!


Implicit Functions

A function f (xy) = 0 is said to be an implicit function, if y cannot be directly defined as a function of x.

In such a case, we will differentiate both sides of this equation w.r.t x, collect the terms containing
Description: 73007.pngon one side; transfer other terms to the other side and divide by the coefficient ofDescription: 73013.pngto get its value.

Description: 73019.png
GivenDescription: 73025.png
Description: 79316.png 

Parametric Equation

If both x and y are functions of a given independent term ti.e.x = f(t) and y = g(t), then the equations containing this x and y are called parametric equations.


The differential of such an equation is given by

Description: 79322.png 

Description: 73049.png
Description: 73055.png 
Description: 79342.png 
Description: 79348.png 

Logarithmic Differentiation

When a function is expressed in any of the following forms, its derivative can be obtained by taking the logarithm of the function and then differentiating it.
  • A product of a number of functions
  • When a function is raised to some exponent which is also a function
  • A number of functions are divided

This method of differentiation is known as logarithmic differentiation.

Description: 73061.png
GivenDescription: 73067.png
Description: 73073.png 
Differentiating both sides, we get
Description: 73079.png 
Description: 79381.png 
Description: 79391.png 
Description: 79397.png 

Higher Order Derivatives

Let y = f (x), be a function of xDescription: 73085.pngis called the first derivative of y with respect to x.



The derivative of f ′(x) is called the second derivative of y with respect to x,


i.e.Description: 73091.png

Description: 73097.png
Description: 73103.png
Description: 73109.png 
Similarly the derivative of f ′′(x) is called the third derivative of y with respect to x.
The nth derivative of y with respect to x is given by Description: 73115.png

Geometric Interpretation of the Derivative

Let y = f (x) be a curve as shown below.


Let P(xy) and Description: 73121.png be two neighbouring points. Join these two points and extend it to meet the x axis at point M.


Slope of PQ is given by Description: 73127.png


As Q approaches P, the line QPM becomes the tangent to the curve at P and the angle q approaches y.


Description: 80273.png 


Description: 79506.png 


Hence, the derivative of y with respect to x is the slope of the tangent to the curve y = f (x) at the point P(xy).

Test Your Skills Now!
Take a Quiz now
Reviewer Name