Question

If f is defined by f (x) = x², find f'(2).

Answer 1

Given:  

\[f(x) = x^2\]

We know a  polynomial function is everywhere differentiable. Therefore 

\[f(x)\]  is differentiable at 

\[x = 2\]

\[f'(2) = \lim_{h \to 0} \frac{f(2 + h) - f(2)}{h}\]
\[ \Rightarrow f'(2) = \lim_{h \to 0} \frac{(2 + h )^2 - 2^2}{h}\]
\[ \Rightarrow f'(2) = \lim_{h \to 0} \frac{(4 + h^2 + 4h) - 4}{h}\]
\[ \Rightarrow f'(2) = \lim_{h \to 0} \frac{h (h + 4)}{h}\]
\[ \Rightarrow f'(2) = 4\]