1. ## Derivatives and Limits

This is my first post, so I'm not sure if I'm doing it correctly... but I have various questions that I am having trouble answering, despite how elementary they probably seem.

1. The derivative of the square root of 3x+4. I do not know where to go after I do 1/2(3x+4)^1/2.. if that is even where I am supposed to go.

2. The derivative is (x+1)x^2 / (x-1)^1/2. I need to find when the continuous function f(x) is increasing, which means I have to set it equal to zero, I believe. When I do this, I keep getting that x = 0. But, therefore I can't solve the problem..?

3. The limit as it approaches positive infinity of f(x)= The square root of (x^2-14) / 3-2x.

Sorry, I could not figure out the answers to these problems.. well, actually, I could not figure out the answer to most of the problems assigned. These were just the ones I had no clue on.

2. Originally Posted by catherine18
This is my first post
Welcome to MHF!

so I'm not sure if I'm doing it correctly... but I have various questions that I am having trouble answering, despite how elementary they probably seem.

1. The derivative of the square root of 3x+4. I do not know where to go after I do 1/2(3x+4)^1/2.. if that is even where I am supposed to go.
You can think of this as a composite function (look that up if you don't know what it is), which means you need to use the chain rule. To do that, you must differentiate the outer function (the square root function) and multiply by the derivative of the inside function (the 3x + 4).

Thus, $\frac d{dx} (3x + 4)^{1/2} = \frac 12 (3x + 4)^{-1/2} \cdot 3$, and you can simplify that.

2. The derivative is (x+1)x^2 / (x-1)^1/2. I need to find when the continuous function f(x) is increasing, which means I have to set it equal to zero, I believe. When I do this, I keep getting that x = 0. But, therefore I can't solve the problem..?
Actually, you need to find where the function is zero OR undefined. Thus your solutions would be $x = -1 \text{ or } 0 \text{ or } 1$ (do you see how I got those?).

Now, the function is increasing when the derivative is positive. Thanks to our solutions, we know we need to find out what is the sign of the derivative on the intervals $(- \infty, -1), (-1,0), (0, 1) \text{ and } (1 , \infty)$, which you can do by testing numbers in each interval.

3. The limit as it approaches positive infinity of f(x)= The square root of (x^2-14) / 3-2x.

Sorry, I could not figure out the answers to these problems.. well, actually, I could not figure out the answer to most of the problems assigned. These were just the ones I had no clue on.
I assume you mean $\lim_{x \to \infty} \frac {\sqrt{x^2 - 14}}{3 - 2x}$

Note that $\frac {\sqrt{x^2 - 14}}{3 - 2x} = \frac {\sqrt{x^2(1 - \frac {14}{x^2})}}{3 - 2x} = \frac {|x| \sqrt{1 - \frac {14}{x^2}}}{3 - 2x}$

Think you can take it from here?

3. Yes, wow, thank you! I really appreciate your help.