Chapter 3 – Test 1

Some examples of differentiation-related problems.

1. Find the derivative of $f(x)=\frac{1}{x^2}$ using the definition.

Solution

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\frac{1}{(x+h{)}^2}-\frac{1}{x^2}}{h}$$ $$=\underset{h\to 0}{\lim }\frac{\frac{x^2-(x+h{)}^2}{x^2(x+h{)}^2}}{h}$$ $$=\underset{h\to 0}{\lim }\frac{x^2-(x^2+2xh+h^2)}{h{x}^2(x+h{)}^2}$$ $$=\underset{h\to 0}{\lim }\frac{-2xh-h^2}{h{x}^2(x^2+2xh+h^2)}$$ $$=\underset{h\to 0}{\lim }\frac{-\cancel{h}(2x+h)}{\cancel{h}{x}^2(x^2+2xh+h^2)}$$ $$=\underset{h\to 0}{\lim }\frac{-(2x+\cancel{h})}{x^4+\cancel{2x^{3}h}+\cancel{x^2{h}^2}}$$ $$=-\frac{2x}{x^4}=-\frac{2}{x^3}$$

2. Find the derivative of $f(x)=x^3+6x$ at $x=4$ using the alternative definition.

Solution

$$f^{\prime }(x)=\underset{a\to 4}{\lim }\frac{[x^3+6x]-[4^3+6(4)]}{x-4}$$ $$=\underset{a\to 4}{\lim }\frac{x^3+6x-88}{x-4}$$ $$=\underset{a\to 4}{\lim }\frac{\cancel{(x-4)}(x^2+4x+22)}{\cancel{(x-4)}}$$ $$=4^2+4(4)+22=54$$