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To understand differentiability, let us understand the meaning of the derivative of a function.
Derivative of a Function
Derivative of a function is the instantaneous rate of a change of dependent variable concerning the independent variable.
For example, if we have a function fx=x2+3x+1 , here f(x) is a function with an independent variable x.
Goal:
To calculate the change in f(x) for an infinitesimal change in x, here is the mathematical notion of the above definition.
Let x be a point in the domain of f(x).
Let ∆x be the change in independent variable x.
Change in the function f(x) will be fx+∆x-f(x).
To calculate the rate of change, we need to divide the change in f(x) with ∆x.
As ∆x tends to zero, the change in f(x) for a very small change in x will give the derivative of a function at point x.
Let us define f'(x) as derivative of function f(x).
f’x=fx+∆x-fx ∆x
This is the formal definition of the derivative of a function; the limit of a function will exist only if the right-hand limit and left-hand limit of the function at that point are equal and finite. Let us consider a point x on function f(x).
The right-hand derivative,
f'(x+)=fx+∆x-fx ∆x
The left-hand derivative,
f'(x-)=fx-∆x-fx -∆x
Only if f’x+=f’x- we can say that the limit exists at f(x) which is derivative of f(x) at x.
Suppose a derivative for a function at a point implies that the function has reasonable and equal limits on either side of the point. In that case, the function is clearly defined (hence it is continuous). The function is said to be differentiable at the point.
Concept of Tangent and its Association with Derivability
If a function is said to be differentiable at a point, then there exists a tangent with a unique slope.
The above figure illustrates the change in function y=f(x) from x to x+∆x.
Slope of line in above figure which joins the points
(x, f(x)) and x+∆x,fx+∆x is
fx+∆x-fx∆x
As ∆x tends to zero, the red line joining two points on the function changes to the tangent at the point x on the curve and the derivative of the function at x will be the slope of the tangent at given point x.
Slope of tangent x=a is equal to fa+h-fa h
1.Discuss the differentiability of f(x)=x if x<1
x2 if x≥1 at x=1
Solution:
RHD
f'(1+)=f1+h-f1 h =1+h2-1 h =h2+2h+1-1 h =h+2 =2
LHD
f'(1-)=f1-h-f1 -h =(1-h)-1 -h =-h -h =1
Since L.H.D is not equal to R.H.D, the function is not differentiable at the given point.
2.Discuss the differentiability at x=0 for the function
f(x)=x2 if x≥0
sin sinx if x<0
Solution:
RHD:
f'(0+)=f0+h-f0 h =sin (0+h) -sin 0 h =sin h h =1
LHD:
f'(0-)=f0-f0-h h =sin 0-h2 h =-h2 h =0
Since L.H.D is not equal to R.H.D, the function is not differentiable at the given point.
- Discuss the differentiability of f(x)={x e-1ǀxǀ+1x, x≠0
0 , x=0 at x=0
Solution:
For x>0, x=x → fx=xe-2x
For x<0, x=-x →fx=x
For x=0, fx=0
Now as x→0,
R.H.L at x=0, fx=xe-2x , f(x) =0
L.H.L at x=0, fx=x, f(x) =0
And f0=0
∵f0-=f0=f(0+)
Hence, the function is continuous.
Now, let us check for the differentiability at x=0
L.H.D at x=0, f’x=x=1
R.H.D at x=0, f’x=e-2x+2xe-2x=0
∵ L.H.D ≠ R.H.D
Function is not differentiable at x=0
Differentiability of a composite function :
fgx is differentiable at x=a if f is differentiable at x=g(a) and g(x) is differentiable at
x=a
Let us suppose that f(x) and g(x) are two functions, let h(x) be a new function defined in a way
h=fog
The function h(x) will be differentiable only if both the functions f(x) and g(x) are differentiable. If either of the functions is/are not differentiable, then h(x) may not be differentiable.
Conclusion
Differentiability of a function at a point x determines how fast the function is changing for an infinitesimal change in x.We also derived formulas for right hand limit and left hand limit.A function is said to be differentiable only if right hand limit is equal to left hand limit.We also took some examples to clear our concepts.
