The chain rule states that
\(\dfrac{\mathrm dw}{\mathrm dt}=\dfrac{\partial w}{\partial x}\dfrac{\mathrm dx}{\mathrm dt}+\dfrac{\partial w}{\partial y}\dfrac{\mathrm dy}{\mathrm dt}+\dfrac{\partial w}{\partial z}\dfrac{\mathrm dz}{\mathrm dt}\)
Since \(\ln(x^2+y^2+z^2)^{1/2}=\dfrac12\ln(x^2+y^2+z^2)\), you have
\(\dfrac{\partial w}{\partial x}=\dfrac x{x^2+y^2+z^2}\)
\(\dfrac{\partial w}{\partial x}=\dfrac y{x^2+y^2+z^2}\)
\(\dfrac{\partial w}{\partial z}=\dfrac z{x^2+y^2+z^2}\)
and
\(\dfrac{\mathrm dx}{\mathrm dt}=\cos t\)
\(\dfrac{\mathrm dy}{\mathrm dt}=-\sin t\)
\(\dfrac{\mathrm dz}{\mathrm dt}=\sec^2t\)
Also, since \(x^2+y^2+z^2=\sin^2t+\cos^2t+\tan^2t=1+\tan^2t=\sec^2t\), the derivative is
\(\dfrac{\mathrm dw}{\mathrm dt}=\dfrac{\sin t\cos t}{\sec^2t}-\dfrac{\sin t\cos t}{\sec^2t}+\dfrac{\tan t\sec^2t}{\sec^2t}\)
\(\dfrac{\mathrm dw}{\mathrm dt}=\tan t\)
