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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.
 

Example
Description: 73019.png
Solution
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 
 

Example
Description: 73049.png
Solution
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.
 

Example
Description: 73061.png
Solution
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
 

Example
Description: 73097.png
Solution
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).





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