Question

Use logarithmic differentiation to find the derivative of the function.

y = (cos 2x)x

Answer 1

y = (cos 2x)^x
log y = x log(cos 2x)
1/y y' = log(cos 2x) + x/cos2x (-2sin2x)

y' = (cos2x)^x (log cos2x - 2x*tan2x)

It is interesting to note that if

y = u^v
y' = vu^(v-1) u' + (logv)(u^v)(v')

If v is a constant, v'=0, so we have the familiar power rule.

If u is a constant, u'=0 and we have the familiar exponentiation rule