Finding the derivative?

Question

Let u(x) = g(f(x)). Find:

a) u' (1)
b) u' (2)
c) u' (3)

Here's a link to the graph

http://www.picturetrail.com/gallery/view?p=999&gid=17982084&uid=9963653&members=1

Answer 1

According to the chain rule, and this works for any compound function:
∂/∂x g( f(x) ) = g'( f(x) ) ⋅ f'(x)

So... u'(1) = ∂/∂x g( f(x=1) ) = g'( f(1) ) ⋅ f'(1)

I believe f(1) is supposed to equal 2... but your graph is drawn poorly. That fact is irrelevant in the entire domain of the functions illustrated anyway
g'(x) = -1, regardless of what its input is.

g'(2) ⋅ 2 = -1 ⋅ 2 = -2

The same process holds for all other inputs.
u'(3) = g'( f(3) ) ⋅ f'(3)
u'(3) = g'( 2 ) ⋅ -2
u'(3) = -1 ⋅ -2
u'(3) = 2

u'(2) = g'( f(2) ) ⋅ f'(2)
u'(2) = g'( 4 ) ⋅ f'(2)
u'(2) = -1 ⋅ f'(2)
The function f(x) is not differentiable at x=2

Answer 2

Note that in the diagram your x axis is not evenly spaced. This is leading to distortions, e.g. f(1) appears to be about 1.5 even though the graph of f has a straight line from (0, 0) to (2, 4). I'm going to assume that the components of f and g are really straight lines, so that f(1) = 2.

If f is differentiable at x and g is differentiable at f(x), then
u'(x) = g'(f(x)) . f'(x) by the chain rule.
So:

a) u'(1) = g'(f(1)) . f'(1)
= g'(2) . f'(1)
= (-1) (2) = -2

b) Doesn't exist since f is not differentiable at 2.

c) u'(3) = g'(f(3)) . f'(3)
= g'(2) . f'(3)
= (-1) . (-2) = 2.