Identify each expression that represents the slope of a tangent to the curve y= -x^3+17x^2-x+3 at any point (x, y).

This is seriously so long I'm not sure it will even fit on a single line. The formula for the derivative using the limiting process is . And of course this is a problem because if h approaches 0, we would have a 0 in the denominator of that fraction and that is definitely not allowed! Every x in the function will be replaced with (x+h) to give us this: . When we expand that we will get this very long numerator (I'm purposely leaving out the limit as h approaches 0 part to save space): . Simplifying that leaves us with this: . We have a lot of terms that cancel each other out so when we do that we are left with . That is one of your choices for answers, the third one down on the left to be specific. Now we can factor out an h: . That h on the top outside the parenthesis cancels with the h on the bottom. Now, as h approaches 0 we have no problems! Yay! That means when we now replace h with 0, we have this: , or simplified we have which is also a choice for your answers, top one on the right. Those are your 2 answers for that dertivative. It's much simpler when you learn the rules!