If f and g are the functions whose graphs are shown, let u(x) = rg(x), (If an an

Question

If f and g are the functions whose graphs are shown, let u(x) = rg(x), (If an answer does not exist, enter DNE.) v(x) = g(Mx)), and w(x) = g(g(x)). Find each derivative, if it exists. If it does not exist, explain why. 0 (a) u1) It does exist. O u'1) does not exist because f (1) does not exist O u'(1) does not exist because g'(1) does not exist. u'(1) does not exist because f '(3) does not exist. O u'(1) does not exist because g'(2) does not exist. It does exist. O v(1) does not exist because f (1) does not exist. exist. O v,(1) does not exist because f(3) does not exist. v'(1) does not exist because g'(2) does not exist.

Explanation / Answer

slope of f(x) from x = 0 to x = 2 is 2 ; (we get slope by 2 point form of line (0 , 0) , (2 , 4) = (4 - 0)/(2 - 0) = 2)

slope of f(x) from x = 2 to x = 7 is -1/4 ; ( (2 , 4) , (6 , 3) = (3 - 4)/(6 - 2) = -1/4 )

slope of g(x) from x = 0 to x = 2 is -3 ; (0 , 6) , (2 , 0) = (0 - 6)/(2 - 0) = -3)

slope of g(x) from x = 2 to x = 5 is 2/3 ; ( (2 , 0) , (5 , 2) = (2 - 0)/(5 - 2) = 2/3)

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

==> u '(x) = f '(g(x)) g '(x) ; chain rule

==> u '(1) = f '(g(1)) g '(1)

==> u '(1) = f '(3) (-3)

==> u '(1) = (-1/4)(-3) = 3/4

Hence u '(1) = 3/4

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

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

==> v'(1) = g '(f(1)) f '(1)

==> v '(1) = g '(2) (2)

since g '(2) is undefined ==> v '(1) is undefined.

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

==> w'(x) = g '(g(x)) g '(x)

==> w '(1) = g '(g(1)) g '(1)

==> w '(1) = g '(3) (-3)

==> w '(1) = (2/3)(-3) = -2

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