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

By definition, the derivative of f(x) is

`lim_(h\rightarrow 0)(f(x+h)-f(x))/(h)`

Let's use the definition for
`f(x)=(1)/(x)`

`lim_(h\rightarrow 0) ((1)/(x+h)-(1)/(x))/(h)=\\lim_(h\rightarrow 0) ((x-(x+h))/(x(x+h)))/(h)=\\lim_(h\rightarrow 0) (((-1)h)/(x^2+xh))/(h)=\\lim_(h\rightarrow 0) ((-1)h)/(h(x^2+xh))=\\lim_(h\rightarrow 0) (-1)/(x^2+xh))=(-1)/(x^2+x*0)=(-1)/(x^2)`

Then,
`f'(x)=(-1)/(x^2)`