Let r = r ( t ) and θ = θ ( t ) with r ( t ) &gt; 0. Let x ( t ) = r ( t ) cos ⁡ ( θ ( t ) ) and y ( t ) = r ( t ) sin ⁡ ( θ ( t ) ) . Prove that d θ d t = 1 x 2 + y 2 ( x d y d t − y d x d t )The hint is to use y ( t ) / x ( t ) and use implicit differentiation but I can't see how to use that hint to solve this problem.

Question

Let   r  =  r  (  t  ) and   θ  =  θ  (  t  ) with   r  (  t  )  &amp;gt;  0. Let   x  (  t  )  =  r  (  t  )  cos  ⁡  (  θ  (  t  )  ) and   y  (  t  )  =  r  (  t  )  sin  ⁡  (  θ  (  t  )  ) . Prove that             d      θ              d      t        =      1                  x        2            +              y        2              (  x            d      y              d      t        −  y            d      x              d      t        )The hint is to use   y  (  t  )      /    x  (  t  ) and use implicit differentiation but I can&#039;t see how to use that hint to solve this problem.

Tristan Pittman · Accepted Answer

We have            y      (      t      )              x      (      t      )        =  tan  ⁡  (  θ  (  t  )  )  .Now differentiate both sides with respect to   t. On the left-hand side, you can differentiate   y  (  t  )      /    x  (  t  ) using the quotient rule. But on the other side we need to use the chain rule. We have            d                      d            t        tan  ⁡  (  θ  (  t  )  )  =      1                  cos        2            ⁡      (      θ      (      t      )      )        ×                    d            θ                      d            t        .Now rearrange for                     d            θ                      d            t      .

Let r=r(t) and theta=theta(t) with r(t)>0. Let x(t)=r(t)cos(theta(t)) and y(t)=r(t)sin(theta(t)) . Prove that (d theta)/(dt)=1/(x^2+y^2)(x(dy)/(dt)−y(dx)/(dt))

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