What's derivative of y=3^(cos(4x-1))

and y=xe^-x - (1/2x^3)^2

and y=sec^2 * (tan(3^x))

The first one, you can use logs to drop down the exponent and then use implicit differentiation to solve it. You'll need to do the product rule and know the rule for derivative of logs.

or

you can just use the rule for exponents

d/dx[ a^x ] = a^x * ln[a]

Either way, you should get the same answer and if it's not the same, you should be able to make them look the same with simple algebra.

the second one, you have to use the product rule for the first part which is [ xe^-x ] and the chain rule for the second part which is (1/2x^3)^2, though i would simplify it a little first.

if the last problem is sec^2 x * tan(3^x) then you have to use the product rule. if the problem is sec^2(tan(3^x)) then you have to do the chain rule multiple times.

y=3^(cos(4x-1))?
dy/dx = 3^(cos(4x-1)) ln(3) (-sin(4x-1) )(4)

and y=xe^-x - (1/2x^3)^2
dy/dx = e^(-x) - x e^(-x) - (1/4) 6 x^5

and y=sec^2 * (tan(3^x))
doesn't make sense