If f is a differentiable function, find an expression for the derivative of each of the following functions.

$($a$)$

y $=$ x$4$f$($x$)$

y$\text{'}=$ x$3$f$\text{'}($x$)3$x$4$f$($x$)$

y$\text{'}=$ x$4$f$($x$)+4$x$3$f$\text{'}($x$)$

$$

y$\text{'}=$ x$4$f$($x$)4$x$3$f$\text{'}($x$)$

y$\text{'}=$ x$4$f$\text{'}($x$)+4$x$3$f$($x$)$

y$\text{'}=$ x$3$f$\text{'}($x$)+3$x$4$f$($x$)$

$($b$)$

y $=$

f$($x$)$x$6$

y$\text{'}=$

xf$\text{'}($x$)6$f$($x$)$x$7$

y$\text{'}=$

xf$\text{'}($x$)5$f$($x$)$x$7$

$$

y$\text{'}=$

xf$($x$)6$f$\text{'}($x$)$x$5$

y$\text{'}=$

xf$($x$)+6$f$\text{'}($x$)$x$6$

y$\text{'}=$

xf$\text{'}($x$)+5$f$($x$)$x$6$

$($c$)$

y $=$

x$7$f$($x$)$

y$\text{'}=$

f$\text{'}($x$)(7$x$6)$ x$7$f$($x$)[$f$($x$)]2$

y$\text{'}=$

f$\text{'}($x$)(6$x$7)+$ x$6$f$($x$)[$f$($x$)]2$

$$

y$\text{'}=$

f$($x$)(7$x$6)$ x$7$f$\text{'}($x$)[$f$($x$)]2$

y$\text{'}=$

f$\text{'}($x$)(6$x$6)$ x$7$f$\text{'}($x$)[$f$($x$)]2$

y$\text{'}=$

f$($x$)(6$x$7)+$ x$7$f$\text{'}($x$)[$f$($x$)]2$

$($d$)$

y $=$

$1+$ xf$($x$)$

x

y$\text{'}=$

xf$\text{'}($x$)+$ x$2$f$($x$)22$x$3/2$

y$\text{'}=$

xf$($x$)+$ x$2$f$($x$)22$x$3/2$

$$

y$\text{'}=$

xf$($x$)+2$x$2$f$\text{'}($x$)12$x$3/2$

y$\text{'}=$

xf$\text{'}($x$)2$x$2$f$\text{'}($x$)+12$x$3/2$

y$\text{'}=$

xf$($x$)$ x$2$f$\text{'}($x$)+12$x$3/2$