Finding derivitive of y with respect to independent variable?

Question

Of the problem y=ln(cos(ln theta))

Answer 1

Use the chain rule

Let y=ln(u), u=cos(w), w=ln(theta)

So y = ln(cos(ln(theta)))

The chain rule says

dy/dtheta = dy/du du/dw dw/dtheta

y=ln(u), u=cos(w), w=ln(theta)

dy/du = 1/u du/dw = -sin(w) dw/dtheta = 1/theta

dy/dtheta = 1/u -sin(w) 1/theta

dy/dtheta = 1/cos(w) -sin(w) 1/theta

dy/dtheta = -sin(ln(theta))/cos(ln(theta)) 1/theta

dy/dtheta = -tan(ln(theta))/theta

Answer 2

1/cos(ln theta) * [-sin(ln theta)] * 1/theta = -tan(ln theta)/theta