derivative of unknown function

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May 18th 2008, 11:50 AM
LeoBloom.

derivative of unknown function

Says here, a function f is defined for all real numbers and has the following property: f(a + b) - f(a) = 3(a^2)b + 2b^2. f ' (x) is equal to what?

Basically, are a and b constants or something? How do I find f ' (x)?
May 18th 2008, 12:08 PM
Soroban

Hello, LeoBloom!

Quote:

A function $f$ is defined for all real $x$ and: . $f(a + b) - f(a) \;= \;3a^2b + 2b^2$

Find $f'(x)$

Recall the defintion: . $f'(x) \;=\;\lim_{h\to0}\frac{f(x+h) - f(x)}{h}$

They have given us about half of this defintion: . $f(a+b) - f(a) \:=\:3a^2b - 2b^2$

Divide by $b\!:\;\;\frac{f(a+b) - f(a)}{b} \;=\;\frac{3a^2b + 2b^2}{b} \;=\;3a^2 + 2b$

Take the limit as $b\to0$

. . $\underbrace{\lim_{b\to0}\frac{f(a+b) - f(a)}{b}}_{\text{This is }f'(a)} \;=\;\lim_{b\to0}(3a^2 + 2b) \;=\;3a^2$

So we have: . $f'(a) \:=\:3a^2$

Therefore: . $\boxed{f'(x) \:=\:3x^2}$
May 18th 2008, 01:27 PM
LeoBloom.

Wow thanks alot, I can't believe I didn't notice how similar it looked to the definition of a derivative.