Application Of Derivative

Derivatives of implicit functions:

Example:
Find the expression for dydx if y⁴ + x⁵ − 7x² − 5x^-1 = 0.

Solution:

Note:In case of implicit function y is not expressed explicitly in terms of x only.

geometrical interpretation of the derivative:

  is nothing but rate of increase of y relative to x.(change in x is very very small).e.g. Velocity and Acceleration in rectilinear motion are rate of change of displacement and velocity w.r.t time respectively.

Slope of tangents and normals:

Let y=f(x) a continuous curve and let    a point on it.

Slope of tangent to the curve at P is

Slope of normal to the curve at P is

increasing and decreasing functions:

Strictly increasing function:

                         For   if    

So f(x) is strictly increasing if  for all x

Strictly decreasing function:

similarly for  for all x

Note:  If   f is decreasing at c

           If   f is increasing at c

maximum and minimum values of a function:

(i)First derivative test:Find critical points(c) of f(x) by solving for points(x) where  or  does not exists.

(ii)Second derivative test:

If   ,f has maximum value(f(c)) at x=c

If  , f has minimum value(f(c)) at x=c

Note: (1)If in second derivative test  ,then find  

If    then f(x) has neither maximum nor minimum(inflexion point) at x=c.

Else if  then

f(x) minimum at x=c

f(x) maximum at x=c

Repeat till the point is discussed.

(2)For absolute maximum or absolute minimum in a closed interval  consider the values of function at end points(a,b) besides extreme points.

 = max {f(a),local maximas,f(b)}=absolute maximum

 = min {f(a),local minima,f(b)}=absolue minimum

Rolle's Theorem:

If f is

(i)continuous in [a,b]

(ii)differentiable in (a,b)

(iii)f(a)=f(b)

then there exist at least a point c (a<c<b) such that 

Geometrically there exist at least a point(c,f(c)) on the curve at which the tangent is parallel to x axis

Lagrange's Mean Value Theorem:

If f is

(i)continuous in [a,b]

(ii)differentiable in (a,b)

then there exist at least a point c (a<c<b) such that

Geometrically there exist at least a point(c,f(c)) on the curve at which the tangent is parallel to chord joining (a,f(a)),(b,f(b)).

Cauchy's mean value theorem:

If f and g are

(i)continuous in [a,b]

(ii)differentiable in (a,b)

(iii)g'(x) is not equal to zero

then there exist at least a point c (a<c<b) such that