Question

Use the algebraic procedure explained in section 8.9 in your book to find the derivative of f(x)=1/x. Use h for the small number. (Hint: Simplify f(x+h)-f(x) by finding a common denominator and combining the two fractions).

Answer 1

Answer:

By definition, the derivative of f(x) is

$lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

Let's use the definition for $f(x)=\frac{1}{x}$

$lim_{h\rightarrow 0} \frac{\frac{1}{x+h}-\frac{1}{x}}{h}=\\lim_{h\rightarrow 0} \frac{\frac{x-(x+h)}{x(x+h)}}{h}=\\lim_{h\rightarrow 0} \frac{\frac{(-1)h}{x^2+xh}}{h}=\\lim_{h\rightarrow 0} \frac{(-1)h}{h(x^2+xh)}=\\lim_{h\rightarrow 0} \frac{-1}{x^2+xh)}=\frac{-1}{x^2+x*0}=\frac{-1}{x^2}$

Then, $f'(x)=\frac{-1}{x^2}$