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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).`
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.
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