Help Solving Equation of Tangent Line (VERY Simple)

[Chris686]

I am jumping back into some Calculus, and I've forgotten some basic algebra. I'm sure this is super easy, but I don't remember what the steps to take are, or what this particular problem would be called.

I'm just trying to simplify this expression:



The book says the first part simplifies to the second part, and I simply don't remember how to do this. In addition, if there were a link someone knew about listing these basics rules, it'd be much appreciated.

 

[galactus]

I moved this thread to Intermediate/Advanced Algebra, though it appears to be the beginnings of the definition of a derivative.

This is an exercise in simplifying rational expressions.

We have \(\displaystyle \displaystyle \frac{\frac{2}{2+h}-1}{h}\)

Now, as with any fraction one is adding or subtracting, you can find the common denominator.

The common denominator is 2+h. We have to write it so a 2+h appears in the denominator of the 1 term. So, multiply top and bottom of 1 by 2+h

Then, we have \(\displaystyle \displaystyle \frac{\frac{2}{2+h}-\frac{2+h}{2+h}}{h}\)

\(\displaystyle \displaystyle =\frac{\frac{2-(2+h)}{2+h}}{h}\)

But, \(\displaystyle \displaystyle 2-(2+h)=-h\). So, it simplifies to:

\(\displaystyle \displaystyle =\frac{-h}{h(2+h)}=\frac{-1}{2+h}\)

 

[Chris686]

Thank you very much. It makes sense now, and I suppose it's better suited for the algebra forum.

 

[Chris686]

Moving along, slowly but surely. I'm doing my homework, and I'm wondering what this means. The book doesn't specify what the difference is. This is what these problems are asking for:

(b) Find the instantaneous rate of change of y with respect to x
at the specified value of x0.
(c) Find the instantaneous rate of change of y with respect to x
at an arbitrary value of x0.

I understand what B is asking for, but any idea what it means by an arbitrary value of x0?

 

[HallsofIvy]

Moving along, slowly but surely. I'm doing my homework, and I'm wondering what this means. The book doesn't specify what the difference is. This is what these problems are asking for:

(b) Find the instantaneous rate of change of y with respect to x
at the specified value of x0.
(c) Find the instantaneous rate of change of y with respect to x
at an arbitrary value of x0.

I understand what B is asking for, but any idea what it means by an arbitrary value of x0?

I assume you are given some specific function, y. Just replace the "x" in the derivative with, literally "\(\displaystyle x_0\)".