Quiz 3 - Definition of a derivative

Problem 1

Question

Use the definition of a derivative to find the derivative $f^{\prime }(x)$ of the function $f(x)=\frac{1}{x}$.

Solution

Using the definition: $$\begin{array}{rl}{f}^{\prime }(x)& =\underset{h\to 0}{lim}\left(\frac{f(x+h)-f(x)}{h}\right)\\ & =\underset{h\to 0}{lim}\left(\frac{\frac{1}{x+h}-\frac{1}{x}}{h}\right)\\ & =\underset{h\to 0}{lim}\left(\frac{\frac{x}{x(x+h)}-\frac{x+h}{x(x+h)}}{h}\right)\\ & =\underset{h\to 0}{lim}\left(\frac{\frac{x-(x+h)}{x(x+h)}}{h}\right)\\ & =\underset{h\to 0}{lim}\left(\frac{\frac{-h}{x(x+h)}}{h}\right)\\ & =\underset{h\to 0}{lim}\left(\frac{-1}{x(x+h)}\right)\\ \boxed{{\displaystyle f^{\prime }(x)=-\frac{1}{x^2}}}\phantom{f^{\prime }(x)}& \phantom{=-\frac{1}{x^2}}\end{array}$$