FIND THE DERIVATIVES FROM THE LEFT AND RIGHT AT THE GIVEN POINT

About "Find the Derivatives From the Left and Right at the Given Point"

Find the Derivatives From the Left and Right at the Given Point :

Here we are going to see how to find the derivatives from the left and right at the given point.

Find the Derivatives From the Left and Right at the Given Point - Examples

For a function y = f(x) defined in an open interval (a, b) containing the point x₀, the left hand and right hand derivatives of f at x = h are respectively denoted by f'(h^-) and f'(h⁺)

f'(h^-)  =  lim _{h-> 0}^-[f(x + h) - f(x)] / h

f'(h⁺)  =  lim _{h-> 0}⁺[f(x + h) - f(x)] / h

provided the limits exist.

Question 1 :

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?

(i)  f(x)  =  |x - 1|

Solution :

If the function is differentiable, then

f'(1^-)  =  f'(1⁺)

f'(1^-)  = lim_x->1- [f(x) - f(1)] / (x - 1)

=  lim_x->1- [-(x - 1) - 0]/(x - 1)

=  -1

f'(1⁺)  =  lim_x->1+  [f(x) - f(1)] / (x - 1)

=   lim_x->1+ [(x - 1) - 0]/(x - 1)

=  1

Hence the given function is not differentiable at x = 1.

(ii)  f(x)  =  √(1 - x²)

Solution :

If the function is differentiable, then

f'(1^-)  =  f'(1⁺)

f'(1^-)  =  [f(x) - f(1)] / (x - 1)

=   lim_x->1- [√(1 - x²) - 0]/(x - 1)

=   lim_x->1- [√(1 - x²) - 0]/(1 - x)

=   lim_x->1- -√(1 + x) / √(1 - x)

=  -√2 / 0

=  -∞

Hence the given function is not differentiable at x = 1.

Solution :

If the function is differentiable, then

f'(1^-)  =  f'(1⁺)

f'(1^-)  = lim_x->1- [f(x) - f(1)] / (x - 1)

=  lim_x->1- (x - 1)/(x - 1)

=  1

f'(1⁺)  =  lim_x->1+  [f(x) - f(1)] / (x - 1)

=   lim_x->1+ (x² - 1)/(x - 1)

=  lim_x->1+ (x + 1)(x - 1)/(x - 1)

=  lim_x->1+ (x + 1)

=  2

f'(1^-)  =  1 and f'(1⁺)  =  2,  so the given function is not differentiable at x = 1.

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