Slope of Tangent line is driving me nuts kind of question

[irishpump]

I recently took an exam, and one of the questions I got wrong was:

The slope of the secant line to the graph of g(x) between x=5 and x=5+h is 10 - 6h + 2h^2. What is the slope of the tangent line at x=5?

I did the 4 step process for the definition of a derivative. He told me its easier than that when I tried to show him the work. I keep looking at this and I cant understand how to do it. He said I need to think about the definition of a derivative and relationship between a secant and tangent line. So since the full 4 step process was unecessary for this kind of problem, I've since tried taking the limit of g(x) approaching 0 and I've tried it approaching 5. I've now gotten the answer of 6 but I have no idea how I can check this, and quite frankly I need some help for this kind of question coming up at the end semester finals.

Any help is very much appreciate!

 

[tkhunny]

The "four step process"? Do tell.

Slope of secant: \(\displaystyle \frac{g(5+h)-g(5)}{h}\;=\;10-6h+2h^{2}\)

It may be confusing that we are NOT given a definition of g(x). We have been given the result of the secant slope.

The tangent slope is defined in terms of the secant slope as the limit as h approaches zero (0). This makes the slope of the tangent 10.

I'd still like to see the "four step process".

 

[irishpump]

Thank you very much! I noticed with the mess I made on my paper that was one of my answers when I set h to zero. The 4 step process as my Professor calls it, really is just the definition of a derivative broken down. So he would have it as:

Step 1: f(x)= X+h
Step 2: f(x)= (X+h) - f(x)
Step 3: f(x)= (X+h) - f(x)/h
Step 4: lim h->0

I don't know if he taught it to this way to make it easier to remember by breaking it down, but I'll be honest and say after awhile I quit labeling each step and just wrote out the entire definition to solve a problem. Thank you again. I kept trying to reintroduce the equation back into the definition of a derivative. I had no idea the result was given, and just simply needed to plug in zero. I know out of frustration I did it at some point of my messy paper lol. Thank you again very much!

 

[tkhunny]

Oddly, you seem not to have made the connection between the process and the secant. Oh well. Some methods work better than others. This one seems a bit excessive, to me.