Thread: Derivation using the definition of a derivite

The question I am struggling with is as follows:

My working is as follows:

I don't know how to continue from here or even rather if i'm on the right track.
Thank you in advance for any help!

Originally Posted by ImPoob

The question I am struggling with is as follows:

My working is as follows:

I don't know how to continue from here or even rather if i'm on the right track.
Thank you in advance for any help!

You are making it entirely too difficult. First of all, you are asked to find $F'(0)$ so $x=0$ and you might better start with$$
F'(0) = \lim_{h\to 0} \frac{F(0+h) - F(0)}{h} = \lim_{h\to 0} \frac {f(h)\sin^2(h)}{h}
$$where I have omitted terms that are $0$. Now think about breaking that into the product of three factors, each of which is easy to find the limit.

Thanks for you help! After mulling over your explanation I understand how you arrived at:


When you say break it into a product of three factors I assume you simply mean:


If this is correct I still wouldn't know how to continue from here, as in all the examples iv'e come across before the h in the denominator cancels. Currently we would be dividing by zero if we took the limit of the function. My only other thought was to rewrite sin^2 (x) in some other fashion using trig identities but that still doesn't help with the h in the denominator.

Thanks a bunch again.

No, that is NOT what he meant! For one thing you have mysteriously changed two of the "h"s to "x"s. Also you did not completely change it into a product of three terms.

You should have $\displaystyle \left(\lim_{h\to 0} f(h)\right)\left(\lim_{h\to 0} sin(h)\right)\left(\lim_{h\to 0}\frac{sin(h)}{h}\right)$.

Ah. Thank you for your help also!

Would the final step be to use limit laws to evaluate the expression using lim h-> 0 sin(h)/h = 1?

Thank again.

Originally Posted by ImPoob

Ah. Thank you for your help also!

Would the final step be to use limit laws to evaluate the expression using $\displaystyle \mathop {\lim }\limits_{h \to 0} \frac{{\sin (h)}}{h} = 1$?

Yes indeed!

And what answer did you get for this problem?