Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. (Simplify your answers completely.) f(x) = −x2 + 6x

Question

f(x) = −x² + 6x

Step 1: f(x +h) =

Step 2: f(x + h) - f(x) =

Step 3: f(x + h) - f(x) / h =

Step 4: f ' (x) = lim h→0 f(x + h) − f(x) / h =

Answer 1

#1: f(x+h) = -(x+h)² + 6(x+h). Simplify. Careful with the negative sign. It's applied after the square.

#2: Subtract -x² + 6x from the result of #1. Watch the signs.

#3: Divide the result of #2 by h. Note that the numerator is everything to the left of the division sign. Simplify.

#4: What happens to the equation from #3 when h becomes vanishingly small, effectively 0? Note that the answer is deceptively simple.