Differentiation of Exponential Functions

Derivatives are most propably used to solve an equation by the application of some of the simple properties.

So, let me explain this statement by a simple illustration

If y = sin x which is a trignometric funtion then it's derivative can be taken as y' =cos x .                                                                                                             In this way we can solve the various homogeneous functions with a simple illustration which ought  to implement  various formulae.the few among them are as follows

d(xn)/dx =nxn-1
d(ex)/dx =ex
d(log x)/dx =1? x



The uv theorem of differentiation is applicable only when the given two functions are of different functions like one is of logarthmic and one is of algebraic function.The application of differentiation is mainly used in calculus specially to find out the rate of change

Differential and derivatives of exponential functions.

PARTIAL DIFFERENTIATION

Partial differentiation arise in variety of problems in science and engineering usually the independent variables are scalars for example,pressure, temperature, density, velocity, force ect.To formulate the partial differential equation from the given physical problem and to solve the mathematical problem.

DERIVATIVES OF TRIGNOMETRIC FUNCTIONS

The various trignometric functions like sin x ,cos x, tan x, cot x, sec x, cosec x can all be solved easily with application of derivatives as shown in the first illustration.

The derivative of an odd function is always even.
IF y=f(x) is a homogenous function of degree n in x then the relative error in y is n times the relative error in x.
The change in y is represented by ?y and the change in is represented by ?x.

DERIVATIVES OF HYPERBOLIC FUNTIONS

The hyperbolic funtions of trignomety as sin hx, cos hx, tan hx, cot hx, sec hx,cosec hx can all be implemented with the application of derivatives to solve the problem in few steps and in a simple way.

SOLVED PROBLEMS

x sin x

sol:         u(x)=x   v(x)= sin x

d ? dx  uv( x ) = u(x) d ? dx v(x) + v(x) d ? dx u(x)

= x d ? dx sin x + sin x dx ? dx

=x cos x +sin x

2. log (sin-1 (ex ))

sol:  u =ex    v = sin-1 u    y = log v

=d(u) ?dx ×d (v) ? dx ×dy ? dx

=ex × 1 ? ?1-u 2 × 1? v

=ex ? sin-1 (ex )?1-e2x

3. tan (ex )

sol:       d ? dx tan ( ex ) =sec2 (ex )  d ? dx (ex )

=ex sec2 (ex )


Algebra is widely used in day to day activities watch out for my forthcoming posts on answers for algebra 2 problems and cbse syllabus for class 9. I am sure they will be helpful.

Differentiation of Exponential Functions-Problems .

4) y  =  e2x log x

derivative is done in the following ways.

y'  =     e2x log x  d(2x log x)

y' =  e2x log x   [log x  d(2x)  + 2x d(log x)]

y' = e2x log x   [2 log x  + 2x/x]

y' =e2x log x  [ 2log x + 2]  Answer.