Definition & Proving Differentiability: A Function f at a Point a

In summary, the definition of a function f being differentiable at a ∈ R is that the limit as x approaches a of {f(x) - f(a)}/(x - a) exists. If f is differentiable at a, then it is also continuous at a. To calculate the derivative of g(x) = (f(x))^0.25, we can use the definition of the derivative and standard limit laws to get 0.25(f(x))^(-0.75)f '(x).
  • #1
Unusualskill
35
1
(a) State precisely the definition of: a function f is differentiable at a ∈ R.

(b) Prove that, if f is differentiable at a, then f is continuous at a. You may
assume that
f '(a) = lim {f(x) - f(a)}/(x - a)
x→a

(c) Assume that a function f is differentiable at each x∈ R and also f(x) > 0
for all x ∈R. Use the definition of the derivative and standard limit laws to
calculate the derivative of:
g(x) = (f(x))^0.25
in terms of f(x) and f '(x).

I did part a n b . But stuck at part c , can any1 guide me on part (c)?thank you
 
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  • #2
To save writing, let u = f(x). Therefore you want d/dx(u^0.25)={d/du(u^0.25)}{du/dx}=0.25u^(-0.75)u'.
 
  • #3
This same question was posted in the "Calculus and Beyond Homework" section and answered there. Unusualsikill, do not post the same thing in more than one section. If a homework section is appropriate, post there only.
 

Related to Definition & Proving Differentiability: A Function f at a Point a

1. What is the definition of differentiability?

The definition of differentiability for a function f at a point a is that the limit of the difference quotient (f(x) - f(a)) / (x - a) exists as x approaches a.

2. How is differentiability different from continuity?

Differentiability is a stronger condition than continuity. A function can be continuous at a point without being differentiable at that point, but a function must be differentiable at a point in order to be continuous at that point.

3. Can a function be differentiable at a point but not continuous?

No, a function must be continuous at a point in order to be differentiable at that point.

4. How can we prove that a function is differentiable at a point a?

To prove differentiability at a point a, we need to show that the limit of the difference quotient (f(x) - f(a)) / (x - a) exists as x approaches a. This can be done by showing that the left and right hand limits of the difference quotient are equal, or by using the definition of the derivative at a point.

5. Are there any functions that are not differentiable at any point?

Yes, there are some functions that are not differentiable at any point, such as the absolute value function. These functions have sharp corners or cusps, which make it impossible for the limit of the difference quotient to exist at those points.

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