Prove f(x) is differentiable at x=1 where f(x)=2x^2 x=<1, =4x-1 x>1

1. Nov 11, 2007

sara_87

prove f(x) is differentiable at x=1:
f(x)=2x^2 x(less than or equal to)1
4x-1 x>1

2. Nov 11, 2007

PhY

do you mean:
d/dx:= d/dx(2x^2x) * d/dx(2x) ala function of a function?

i'm probably wrong.

3. Nov 11, 2007

sara_87

no, i mean it's in piecewise form with the big curly brackets

4. Nov 11, 2007

PhY

You Mean Integration.

Argh, My Integration is Rusty.
its the opposite of Differentiation.

so 2x would be x^2
2x^2x ...i don't know, because its F(X)^G(X).

You need to hear from somebody else on this.

5. Nov 11, 2007

sara_87

yeah i think i do need to hear from somebody else on this cos u didnt understand the question ;)
i have to prove that it is differentiable at x=1 it has nothing to do with integration.
thanks anyway

6. Nov 11, 2007

cristo

Staff Emeritus
The derivative of a function at a point can be expressed as the limit of an expression. You should be able to get two limits; one for each branch of the function. If these are the same, then the function is differentiable at x=1.

7. Nov 11, 2007

JasonRox

It's amazing how complicated one can make a question when it is meant to be simple. (Not meant to the OP.)

8. Nov 11, 2007

sara_87

thanx v much i think i can do it now.

9. Nov 11, 2007

CompuChip

You might also want to check that it is continuous. For example,
$$f(x) = \left\{ \begin{array}{rl} 0 & \text{ if } x \le 0 \\ 1 & \text{ if } x > 0 \end{array}$$
will give you 0 for the derivative when approaching from the left or right to zero, though at x = 0 the function is not continuous at all.

Last edited: Nov 11, 2007
10. Nov 11, 2007

HallsofIvy

Staff Emeritus
No, he meant differentiation. Determine whether
$$f(x) = \left\{ \begin{array}{rl} 2x^2 & \text{ if } x \le 1 \\ 4x-1 & \text{ if } x > 1 \end{array}$$
is differentiable at x= 1

Compuchip's suggestion is etremely good here: a function can't be differentiable if it is't continuous at the point! What is the limit of f as you approach 1 from the left? What is the limit as you approach from the right?

11. Nov 12, 2007

Gib Z

Or more precisely, $$\frac{d}{dx}\int^x_a f(t) dt = f(x)$$ where a is a constant. "The Opposite of differentiation" is what people told me before I started integral calculus as well, and it screwed up my understanding a heap load.

Just in case you want to know, you let y=2x, and express the remaining integral in terms of the exponential and logarithmic functions.

12. Nov 12, 2007

CompuChip

I still don't see why you would want to do integration.
It's a partwise defined function, all you need to do is show that the function and the derivative are continuous.