Question

Find the derivative of f(x) = x squared - 4 by definition

y' = lim f( x+h) - f(x) / h
h approaching 0

Answer 1

Find the derivative of f(x) by definition

$$f(x)=x^2-4$$ $$y'=\lim_{h\to0} \cfrac{f(x+h)-f(x)}{h}$$

Solution:

$$f'(x)=\lim_{h\to0}\cfrac{(x+h)^2-4-x^2+4}{h}$$ $$\iff f'(x)=\lim_{h\to0}\cfrac{x^2+2xh+h^2-x^2}{h}$$ $$\iff f'(x)=\lim_{h\to0}\cfrac{2xh+h^2}{h}$$ $$\iff f'(x)=\lim_{h\to0}\cfrac{h(2x+h)}{h}$$ $$\iff f'(x)=\lim_{h\to0}(2x+h)$$ $$\iff f'(x)=2x$$

Answer:

$$f'(x)=2x$$