# Thread: Derivation using the definition of a derivite

2. ## Re: Derivation using the definition of a derivite Originally Posted by ImPoob 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.

3. ## Re: Derivation using the definition of a derivite

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.

4. ## Re: Derivation using the definition of a derivite

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

5. ## Re: Derivation using the definition of a derivite

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.

6. ## Re: Derivation using the definition of a derivite 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!

7. ## Re: Derivation using the definition of a derivite

And what answer did you get for this problem?

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