SOLUTION: Let {{{y =(f(u) + 4x)^2}}} and {{{u = x^3 - 2x}}}. If {{{f(4) = 7}}} and {{{dy/dx = 16}}} when {{{x = 2}}}, find f '(4).

Question 1161974: Let and . If and when , find f '(4).
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

When x = 2,
u = x^3 - 2x
u = (2)^3 - 2(2)
u = 4
f(u) = f(4) = 7
So x = 2 leads to f(u) = 7

y = ( f(u) + 4x )^2
dy/dx = 2( f(u) + 4x )*( f'(u) + 4 ) ... chain rule
16 = 2( 7 + 4(2) )*( f'(u) + 4 ) ... substitution; isolate f'(u)
16 = 2( 15 )*( f'(u) + 4 )
16 = 30( f'(u) + 4 )
16 = 30*f'(u) + 120
16-120 = 30*f'(u) ... subtract 120 from both sides
-104 = 30*f'(u)
30*f'(u) = -104
f'(u) = -104/30 ... divide both sides by 30
f'(u) = -52/15 ... reduce
f'(4) = -52/15 ... recall that x = 2 leads to u = 4

Answer: -52/15

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