showing function differentiable

**hunkydory19** · February 1st 2008, 04:03 AM

Let [0,1] --->R be continuous. (little f above arrow)
Define [0,1] --->R (capital F above arrow) by F(x) = $\displaystyle\int^{x^2}_0 f \,$

Show that F is differentiable with $F'(x) = 2x f(x^2).$

I seriously have worked on this question for hours and have got nowhere! Can anyone give me any help at all?!

Thanks in advance!

**colby2152** · February 1st 2008, 05:01 AM

Originally Posted by hunkydory19

Let [0,1] --->R be continuous. (little f above arrow)
Define [0,1] --->R (capital F above arrow) by F(x) = $\displaystyle\int^{x^2}_0 f \,$

Show that F is differentiable with $F'(x) = 2x f(x^2).$

I seriously have worked on this question for hours and have got nowhere! Can anyone give me any help at all?!

Thanks in advance!

Let $G(x) = \displaystyle\int^{x^2}_0 f \,$

$G(x)=F(x^2)-F(0)$

$G'(x)=f(x^2)2x-f(0)$ , assume $f(0)=0$ to get: $G'(x)=f(x^2)2x$

I used G instead of F for the left side so you wouldn't confuse the two!

**bobak** · February 1st 2008, 06:20 AM

am i missing something here or did you assume that $F(0) = 0$

**colby2152** · February 1st 2008, 06:22 AM

Originally Posted by bobak

am i missing something here or did you assume that $F(0) = 0$

Yes, I added that assumption in.

**bobak** · February 1st 2008, 06:25 AM

Originally Posted by colby2152

Yes, I added that assumption in.

surely $F(0)$ is constant so its differentiates to give 0

**colby2152** · February 1st 2008, 06:27 AM

Originally Posted by bobak

surely $F(0)$ is constant so its differentiates to give 0

Good point, because $F(0)=\int_0^0 fdx \Rightarrow 0$

**bobak** · February 1st 2008, 06:39 AM

Originally Posted by colby2152

Good point, because $F(0)=\int_0^0 fdx \Rightarrow 0$

isnt that $G(0)$ ?

**colby2152** · February 1st 2008, 06:44 AM

Originally Posted by bobak

isnt that $G(0)$ ?

Yes, in my example, that is $G(0)$

Cite back to my original reply.

**ThePerfectHacker** · February 1st 2008, 07:49 AM

Originally Posted by hunkydory19

Let [0,1] --->R be continuous. (little f above arrow)
Define [0,1] --->R (capital F above arrow) by F(x) = $\displaystyle\int^{x^2}_0 f \,$

Show that F is differentiable with $F'(x) = 2x f(x^2).$

I seriously have worked on this question for hours and have got nowhere! Can anyone give me any help at all?!

Thanks in advance!

Let $\chi (x) = \int_0^x f(x) dx$ for $x\in [0,1]$ .
Let $\mu (x) = \int_0^{x^2} f(x) dx$ for $x\in [0,1]$ .
We wish to show $\mu$ is differenciable and find its derivative.

Note, $\mu (x) = \chi (x^2)$ . Now $\chi$ and $x^2$ are differenciable on $(0,1)$ by the fundamental theorem. This means, $\mu ' (x) = 2x \chi ' (x^2) = 2x f(x^2)$ .

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