Section 1
Rules of Differentiation for a Function of One Variable
Find the derivative of each of the following functions:$$(a) y=x^{12}$$$$(b) y=63$$$$(c) y=7 x^{5}$$$$(d) w=3 u^{-1}$$$$(e) w=-4 u^{1 / 2}$$$$(f) w=4 u^{1 / 4}$$
Find the following:$$(a) \frac{d}{d x}\left(-x^{-4}\right)$$$$(b) \frac{d}{d x} 9 x^{1 / 3}$$$$(c) \frac{d}{d w} 5 w^{4}$$$$(d) \frac{d}{d x} c x^{2}$$$$(e) \frac{d}{d u} a u^{b}$$$$(f) \frac{d}{d u}-a u^{b}$$
Find $f^{\prime}(1)$ and $f^{\prime}(2)$ from the following functions:$$(a) y=f(x)=18 x$$$$(b) y=f(x)=c x^{3}$$$$(c) f(x)=-5 x^{-2}$$$$(d) f(x)=\frac{3}{4} x^{4 / 3}$$$$(e) f(w)=6 w^{1 / 3}$$$$(f) f(w)=-3 w^{-1 / 6}$$
Graph a function $f(x)$ that gives rise to the derivative function $f^{\prime}(x)=0 .$ Then graph a function $g(x)$ characterized by $g^{\prime}\left(x_{0}\right)=0$