Looks like [tex]x(t)=\tan t+3t[/tex] and [tex]y(t)=\frac t{\cos t}=t\sec t[/tex].
By the chain rule,
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}[/tex]
We have
[tex]\dfrac{\mathrm dy}{\mathrm dt}=\sec t+t\sec t\tan t[/tex]
[tex]\dfrac{\mathrm dx}{\mathrm dt}=\sec^2t+3[/tex]
so that the derivative is
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\sec t+t\sec t\tan t}{\sec^2t+3}[/tex]
We can simplify a bit. Multiply the numerator and denominator by [tex]\cos^2t[/tex]:
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\cos t+t\sin t}{1+3\cos^2t}[/tex]
