Derivative of an accumulation function.

[InaudibleTree]

Homework Statement

$F(x) = \int^{ln(x)}_{\pi}cos(e^t)\,dt$

Homework Equations

The Attempt at a Solution

Following from a theorem given in the text I am using:

If f is continuous on an open interval I containing a, then, for every x in the interval,
$d/dx[\int^x_af(t)\,dt] = f(x)$

I thought it would be as simple as

$F'(x) = d/dx[sin(e^{ln(x)})] = d/dx[sin(x)] = cos(x)$

But according to the text the answer is $cos(x)/x$

So what am i doing wrong?

 

[Dick]

Homework Statement

$F(x) = \int^{ln(x)}_{\pi}cos(e^t)\,dt$

Homework Equations

The Attempt at a Solution

Following from a theorem given in the text I am using:

If f is continuous on an open interval I containing a, then, for every x in the interval,
$d/dx[\int^x_af(t)\,dt] = f(x)$

I thought it would be as simple as

$F'(x) = d/dx[sin(e^{ln(x)})] = d/dx[sin(x)] = cos(x)$

But according to the text the answer is $cos(x)/x$

So what am i doing wrong?

It's not quite that simple if your limit isn't simply x. $d/dx[\int^{g(x)}_af(t)\,dt] = f(g(x))g'(x)$. In general you have to use http://en.wikipedia.org/wiki/Leibniz_integral_rule