State both limit definitions of the derivative at 𝑥 = 𝑎.

Answer 1

Our formal definition of a derivative states that

f'(x) = lim_{h-> 0} [f(x+h)-f(x)]/h,

which will give us the derivative as a function of x, or f'(x). If we want the derivative at x=a, or f'(a), all we need to do is replace our x's with a's to get our first answer, ie.

f'(a) = lim_{h-> 0} [f(a+h)-f(a)/h].

For our second answer, we'll compute the definition slightly more directly. Imagine two points on the x-axis, x=a and x=x. These points will each have a corresponding y-value, f(a) and f(x), respectively. To find the slope of the line between these to points, we simply take rise/run, or

[f(x) - f(a)] / [x - a]

We find our derivative at point a, or the slope of the tangent line at point a, by moving x closer and closer toward a, thereby shrinking the interval over which we are finding the slope of the line between the two points. Moving x closer and closer to a is the same as taking the limit as x approaches a, so our derivative at x=a becomes:

f'(a) = lim_{x-> a} [f(x)-f(a)]/[x-a].

I hope that's helpful!