My Notes 30. +0/0.2 points | Previous Answers WebAssignCalcET2 2.6.021a Given th

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My Notes 30. +0/0.2 points | Previous Answers WebAssignCalcET2 2.6.021a Given the graphs of rx) and g(x), estimate u'(2), v 2), and W'(2) if u(x)=rg(x)), v(x)=g(f(x)), and w(x)=g(g(x)). exist, enter DNE.) (If an answer does not u'(2) = w'(2) = 6 g(x) f(x) 4 2 4 6 8 Recall the Chain Rule. If ru) and g(x) are differentiable functions of u and x respectively, then (fo g)'(x) = f'(g(x)) ·g(x). Given a graph of a function y = rx), how can one determine the value ra) for a particular x-value x = a? Given the graph of a piecewise linear function rx), how can one determine f'(a) for a particular x-value x=a? How is f'(g(a)) evaluated? Submit Answer Save Progress Practice Another Version

Explanation / Answer

solution:

using chain rule

a)

u(x)=f(g(x))

u'(x)=f'(g(x))*(g'(x))

now for u'(2)

u'(2)=f'(g(2))*(g'(2))

g'(2) is the slope of g(x) at x=2 which is -8/4= -2

and g(2)=4

so

u'(2)=f'(4)*(-2)

now f'(4) is slope of f(x) at x=4 which is -1

hence

u'(2)=(-1)*(-2)= 2

b)

v(x)=g(f(x))

v'(x)=g'(f(x))*(f'(x))

now for v'(2)

v'(2)=g'(f(2))*(f'(2))

f'(2) is the slope of f(x) at x=2 which is -1

and f(2)=2

so

v'(2)=g'(2)*(-1)

now g'(2) is slope of g(x) at x=2 which is -2

hence

v'(2)=(-2)*(-1)= 2

c)

w(x)=g(g(x))

w'(x)=g'(g(x))*(g'(x))

now for w'(2)

w'(2)=g'(g(2))*(g'(2))

g'(2) is the slope of g(x) at x=2 which is -8/4= -2

and g(2)=4

so

w'(2)=g'(4)*(-2)

now g'(4) is slope of g(x) at x=4 which is 0

hence

w'(2)=(0)*(-2)= 0

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