Estimating a derivative from the limit definition

Question

Estimate f′(3) for f(x)=3x. Be sure your answer is accurate to within 0.1 of the actual value

I don't understand how I'm supposed to do this. I get different answers every time I figure it.

Answer 1

what's the definition of a derivative?

lim_h->0 [ f(x+h) - f(x) ] /h

(aka, the slope of a function at a given point)

plugging in our example → lim_h->0 [ 3*(x+h) - 3(x) ]/h .... we know that 3(x+h) - 3x = 3h so

→lim_h->0 3h/h = 3

the slope should be constant (its the m in y = mx + b for intercept form)

because the question asks for us to estimate this, I expect they want you to calculate the expression

lim_h->0 [ 3*(x+h) - 3(x) ]/h for various values of h.

As h gets smaller, the answer will get closer to 3