Help with using the fact that the limit, as t->x, of (sec(t) - sec(x))/(t - x) = f(x) to find f'((pi)/4)

[cubester04]

I have the following problem,
If [imath]f( x) =\lim _{t\rightarrow x}\frac{\sec t-\sec x}{t-x}[/imath], find the value of [imath]f'\left(\frac{\pi }{4}\right)[/imath].
My first instinct is to use L'Hôpital's rule since if I plug in x for t, I just get 0/0, but that didn't seem to lead anywhere for me. Could someone please provide some suggestions for me on how to get started?

Any help will be appreciated.

 

[fresh_42]

I have the following problem,
If [imath]f( x) =\lim _{t\rightarrow x}\frac{\sec t-\sec x}{t-x}[/imath], find the value of [imath]f'\left(\frac{\pi }{4}\right)[/imath].
My first instinct is to use L'Hôpital's rule since if I plug in x for t, I just get 0/0, but that didn't seem to lead anywhere for me. Could someone please provide some suggestions for me on how to get started?

Any help will be appreciated.

Shouldn't it be [imath] f'(x)=\lim_{t \to x}\dfrac{\sec t -\sec x}{t-x} [/imath] since the RHS is already the derivative?

Are you allowed to use [imath] \dfrac{d}{dx} \sec x =\dfrac{\sin x}{\cos^2x} [/imath]? Otherwise, use the definition of the secant or try [imath] f'(x)=\lim_{h\to 0}\dfrac{\sec (x+h) -\sec x}{h} [/imath]

 

[cubester04]

It's f(x), not f'(x). I'll try your suggestions.

 

[cubester04]

I think I figured it out with the help of a tutor. It's really simple. I probably should have been able to see this.
[math]f( x) \ =\ \lim _{t >x}\frac{sec( t) -sec( x)}{t-x}[/math]
Use l'Hopistal's rule and differentiate with respect to t (x is a constant):

[math]f( x) \ =\lim _{t >x}\frac{sec( t) \ tan( t)}{1} \ =\ \lim _{t >x} \ sec( t) \ tan( t)[/math]
Plug in x for t:

[math]f( x) \ =\ sec( x) \ tan( x)[/math][math]\begin{array}{l} f'( x) \ =\ sec( x) \ sec^{2}( x) \ +\ sec( x) \ tan( x) \ tan( x)\\ \end{array} \\f'( x) \ =\ sec^{3}( x) \ +\ sec( x) \ tan^{2}( x) \\f'\left(\frac{\pi }{4}\right) \ =\ 8\ +\ \sqrt{2\ }[/math]
From what I can tell, this is the solution...

 

[khansaheb]

(x is a constant):

The correct statement should be - 'x' is independent of 't'. Later on you were differentiating w.r.t. "x". You could not do that if 'x' if "x" was truly a constant. Similar situation comes up when we have function of multiple variables and execute "partial differentiation".

 

[Steven G]

f(x) is a derivate of some function because of the way f(x) is defined. f(x) is the derivative of secx which is secxtanx. So f'(x) is simply the derivative of secxtanx.

There is no need for L'Hopital's rule.