If f( w ) = e^− 7 sin w + 2 cos w , find f ' ( w ) . Use exact values.

Answer 1

If I an interpreting this function correctly,

f(w) = e^{-7sin(w) + 2cos(w)}

So we need to apply the rule for differentiation of the exponential function along with the chain rule...

let u(w) = -7sin(w) + 2cos(w)

then f(u) = e^u

f'(w) = df/dw = df/du * du/dw [ chain rule ]

df/du = e^u

du/dw = -7cos(w) - 2sin(w) [ these are the differentiation rules for sinusoidal functions ]

Putting this altogether --> f'(w) = e^{-7sin(w) + 2cos(w)} * ( -7cos(w) - 2sin(w) )

With simplification --> f'(w) = -e^{-7sin(w) + 2cos(w)} * (7cos(w) +2sin(w))