Question 5 Let f(x)=1xf(x) = \frac{1}{x} Use the definition of the derivative to find an expression for f′(x)f'(x). Then evaluate f′(2)f'(2) using your expression. Show solutionHide solution+Question 5 - Solution We are given: f(x)=1xf(x) = \frac{1}{x} Use the definition of the derivative: f′(x)=limh→0f(x+h)−f(x)h=limh→01x+h−1xhf'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{h \to 0} \frac{\frac{1}{x + h} - \frac{1}{x}}{h} Combine the terms in the numerator: =limh→0x−(x+h)x(x+h)h=limh→0−hx(x+h)h= \lim_{h \to 0} \frac{\frac{x - (x + h)}{x(x + h)}}{h} = \lim_{h \to 0} \frac{\frac{-h}{x(x + h)}}{h} Simplify: =limh→0−1x(x+h)= \lim_{h \to 0} \frac{-1}{x(x + h)} Take the limit: f′(x)=−1x2f'(x) = \frac{-1}{x^2} Now evaluate at x=2x = 2: f′(2)=−122=−14f'(2) = \frac{-1}{2^2} = \frac{-1}{4} Final Answer: f′(x)=−1x2,f′(2)=−14f'(x) = \frac{-1}{x^2}, \quad f'(2) = \boxed{-\frac{1}{4}}
