1. Derivative Help

Hey, I am running through some examples trying to get ready for my test and am completely stuck on one. I have a feeling something similar will be on our test too!

Anyways here is the question

Using the definition of the derivative find f ' (x). Then find f ' (-2), f ' (0), and f ' (3).

Problem: f(x) = -3 square root of x

Sorry, I can;t figure out how to make the square root sybol but you get the idea..-3 is on the outside with x inside the square root symbol. I mainly need to know f ' (x) and then obviously finding the others will be pretty easy. Thanks in advance!

2. Originally Posted by Mathamateur
Hey, I am running through some examples trying to get ready for my test and am completely stuck on one. I have a feeling something similar will be on our test too!

Anyways here is the question

Using the definition of the derivative find f ' (x). Then find f ' (-2), f ' (0), and f ' (3).

Problem: f(x) = -3 square root of x

Sorry, I can;t figure out how to make the square root sybol but you get the idea..-3 is on the outside with x inside the square root symbol. I mainly need to know f ' (x) and then obviously finding the others will be pretty easy. Thanks in advance!
Been a while since I did one of these!

$\displaystyle f(x) = -3\sqrt{x}$

$\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

So:
$\displaystyle f'(x) = \lim_{h \to 0} \frac{-3\sqrt{x+h} + 3\sqrt{x}}{h}$

We need to rationalize the numerator:
$\displaystyle f'(x) = \lim_{h \to 0} \frac{-3\sqrt{x+h} + 3\sqrt{x}}{h} \cdot \frac{-3\sqrt{x+h} - 3\sqrt{x}}{-3\sqrt{x+h} - 3\sqrt{x}}$

$\displaystyle f'(x) = \lim_{h \to 0} \frac{9(x + h) - 9x}{h(-3\sqrt{x+h} - 3\sqrt{x})}$

$\displaystyle f'(x) = \lim_{h \to 0} \frac{9x + 9h - 9x}{h(-3\sqrt{x+h} - 3\sqrt{x})}$

$\displaystyle f'(x) = \lim_{h \to 0} \frac{9h}{h(-3\sqrt{x+h} - 3\sqrt{x})} = \lim_{h \to 0} \frac{9}{-3\sqrt{x+h} - 3\sqrt{x}}$

Now take the limit:
$\displaystyle f'(x) = \frac{9}{-3\sqrt{x} - 3\sqrt{x}} = -\frac{9}{2 \cdot 3\sqrt{x}} = -\frac{3}{2\sqrt{x}}$

I'll let you take it from here.

-Dan

PS Why are f'(-2) and f'(0) undefined? Describe what's happening to the function at f(0).