Find the derivative of f (x) = 4.15cos^x

Did you mean the derivative of (cos x) ^x? Then, use logarithmic differentiation:

y = (cos x) ^x

ln (y) = ln[ (cos x) ^x]

ln (y) = x * ln[cos x]

(1/y) * y' = 1 * ln[cos x] + x * (-sin x) / (cos x)

y' / y = ln[cos x] - (x sin x) / (cos x)

y' = y * [ ln[cos x] - (x sin x) / (cos x) ]

the key step in the process - the substitution.

You had: y = (cos x) ^x, so we will substitute for y.

y' = (cos x) ^x * [ ln[cos x] - (x sin x) / (cos x) ]

(and this is your final answer)