DIFFERENTIALITY

DIFFERENTIALITY

Differential means derivative of function exist at the point.
OR
Differential means the functions has not any sharp corner in the given domain. 

Example:



The above function is not deferential at X=0 because the function has sharp corner at X=0. 

How to find the differential of the function:

• Find the differential of the function at the LHS and RHS of the given point.
• This LHS and RHS value of differential must be equal and equal to value of differential at the given point. 

Example: The function f(x) = |x| is not differential at X=0 because the function

LHS Derivative

 f(x) = -x

and derivative of 

f(x)=-1

and 

RHS derivative of 

f(x)=1

and value of function at X=0 is zero.

Hence LHS derivative not equal to RHS derivative, not equal to the value at X=0.

hence the function is not differential at X=0

Derivative of function can be found out by at X=c.

limh→0 f(c+h) − f(c)h

NOTE:
If the function is continuous at the given point and it is not necessary it is differential at the given point.

EXAMPLE:

f(x) = |x| 

is continuous at X=0 but not differential X=0 because at the X=0 the function has sharp corner. 

DERIVATIVE

Derivative means slope of the function.

or

Slope means the rate of change of curve.

So the derivative of x² is 2x 

that is rate of change of y w.r.t. is  2x .

Example: what is the derivative of sin(x) ?

and derivative of  sin(x) is cos(x)

i.e.

that is rate of change of  sin(x) is cos(x)

Derivative of exponential function:

Y=e^x

IS e^x 

AND 

Y=ln(x)

IS 1/X

SOME FUNCTIONS AND THEIR DERIVATIVE:

sin(x) = cos(x)

cos(x)= -sin(x)

                                          tan(x)= sec²(x)

                                        cot(x)=-cosec²(x)

                                                e^{x       =}       e^x   

                                                    log(x)=1∕x