Question #99a0a

1 Answer
Feb 9, 2018

# \ #

# y' \ = \ ( 13 x^4 + 6 ) ( x^4 + 6 )^2 . #

Explanation:

# \ #

# \mbox{The key to this is the product rule first, then the chain rule} \ \ \mbox{following that. Here we go:} #

# \mbox{1) Function:} \qquad \qquad \qquad \qquad \ y \ = \ x ( x^4 + 6 )^3. #

# \mbox{2) Product Rule:} \qquad \qquad y' \ = \ x [ ( x^4 + 6 )^3 ]' \ + \ [ x ]'( x^4 + 6 )^3. #

# \mbox{3) Chain Rule:} \ \ y' \ = \ x [ 3 ( x^4 + 6 )^2 ( x^4 + 6 )' ] + [ 1]( x^4 + 6 )^3. #

# \mbox{4) Power Rule:} \qquad \qquad \quad \ y' \ = \ x [ 3 ( x^4 + 6 )^2 ( 4 x^3 ) ] + ( x^4 + 6 )^3; #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ y' \ = \ 12 x^4 ( x^4 + 6 )^2 + ( x^4 + 6 )^3. #

# \mbox{5) Simplification -- factor out lowest powers same of quantities:} #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ y' \ = \ ( x^4 + 6 )^2 [ 12 x^4 + ( x^4 + 6 )^1 ]. #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ y' \ = \ ( x^4 + 6 )^2 [ 12 x^4 + x^4 + 6 ]. #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ y' \ = \ ( 13 x^4 + 6 ) ( x^4 + 6 )^2 . #

# \mbox{6) Statement of Result:} #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ y \ = \ x ( x^4 + 6 )^3. #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ y' \ = \ ( 13 x^4 + 6 ) ( x^4 + 6 )^2 . #

# \ #

# \mbox{[Optional Exercise:} #

# \mbox{Can you see now that} \ y' \ \mbox{is always positive ?} #

# \mbox{A conclusion is: the graph of this function rises forever. } #

# \mbox{And, in particular:} \qquad \qquad \mbox{ the graph of this function has no turning points.] } #