Answer to Question #270065 in Calculus for Festus

Question

If x = t^3 + y^3 where t= a cos v , y= Vsin v. Find dx/dv

Answer 1

$$dx/dv=dx/dt(dt/dv)+dx/dy(dy/dv)$$

$$=3t^2\big(-a\sin (v)\big)+3y^2\big(\sin (v)+v\cos(v)\big)$$

$$=-3a\big(a\cos(v)\big)^2\sin(v)$$

$$+3\big((v\sin(v)\big)^2\big(\sin(v)+v\cos(v)\big)$$

$$=-3a^3\sin(v)\cos^2(v)+3v^2\sin^2(v)\big(\sin(v)+v\cos(v)\big)$$

Or

$$x(v)=a^3\cos^3(v)+v^3\sin^3(v)$$

$$dx/dv=-3a^3\sin(v)+\cos^2(v)$$

$$+3v^2\sin^3(v)+3v^3\sin^2(v)\cos(v)$$