9514 1404 393
Answer:
u'(1) = 0
v'(5) = -2/3
Step-by-step explanation:
There are several ways to go at this. One is to write the function composition, then differentiate that. Another is to make use of the derivative relations only at the point of interest. (The first method is used in the graph.)
At x=1
u(x) = f(x)g(x)
u'(1) = f'(1)g(1) +f(1)g'(1)
The derivatives of the functions will be their slopes at the point of interest.
f(1) = 2, f'(1) = 2
g(1) = 1, g'(1) = -1
u'(1) = (2)(1) +(2)(-1)
u'(1) = 0
__
At x=5
v(x) = f(x)/g(x)
v'(5) = (g(5)f'(5) -f(5)g'(5))/g(5)²
The relevant function values are ...
f(5) = 3, f'(5) = -1/3
g(5) = 2, g'(5) = 2/3
v'(5) = ((2)(-1/3) -(3)(2/3))/(2²) = (-2/3 -2)/4 = (-8/3)/4
v'(5) = -2/3
_____
Additional comment
The functions used in each of the compositions are only defined on the relevant interval [0, 2] or (2, ∞).
