Question

find y' for y=e^(sin(42x))
use the 3 fold chain rule if you can

Accepted Answer

You should remember that you need to use the chain rule when you should evaluate the derivative of composition of functions such that:
(f(g(x)))' = f'(g(x))*g'(x)*x'
Reasoning by analogy yields:
(dy)/(dx) = (d(e^(sin(42x))))/(dx)
(dy)/(dx) = (d(e^(sin(42x))))/(dx)*(d(sin (42 x)))/(dx)*(d(42x))/(dx)
(dy)/(dx) = e^(sin(42x))*cos(42x)*42
Hence, evaluating the derivative of the given composition of functions, using the chain rule, yields (dy)/(dx) = 42e^(sin(42x))*cos(42x).

Answer

Three fold chain rule can be applied here if we think
y = f(g(h(x))) = e^(Sin(42x))
where, "f" is exponential, "g" is Sine, and "h" is 42*x,
so 
y' = dy/dx = (df/dg)(dg/dh)(dh/dx) ----------------(1)
              
now, f(g) = e^g
so    df/dg = e^g
then, g(h) = Sin(h)
so dg/dh = Cos(h)
finally, h(x) = 42*x
so       dh/dx = 42
Hence,  
using equation no. (1) we get,
y' = e^g * Cos(h) * 42
   = e^(Sin(42x)) * Cos(42x) * 42, is the answer.

Math

find y' for y=e^(sin(42x)) use the 3 fold chain rule if you can

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