Question

How do you use the limit definition to find the slope of the tangent line to the graph `f(x) = x^2`?

Answer 1

`f'(x) = 2x`

Explanation:

By definition `f'(x) = d/dxf(x) = lim_{h->0) (f(x+h)-f(x))/h`

So if `f(x)=x^2` then:

`f'(x) = lim_{h->0) ((x+h)^2-x^2)/h`
`:. f'(x) = lim_{h->0) (x^2+2hx+h^2-x^2)/h`
`:. f'(x) = lim_{h->0) ( 2hx+h^2 )/h`
`:. f'(x) = lim_{h->0) ( 2x+h )`
`:. f'(x) = 2x`