Question

What is the derivative of `f(t) = ((lnt)^2-t, tsint )`?

Answer 1

`(df)/(dx)=(tsint+t^2cost)/(2lnt-t)`

Explanation:

Derivative of a parametric function such as `f(x(t),y(t))` is given by `(dy)/(dx)=((dy)/(dt))/((dx)/(dt))`.

Here as `y(t)=tsint`, `(dy)/(dt)=1xxsint+txxcost=sint+tcost`

and as `x(t)=(lnt)^2-t`, `(dx)/(dt)=2lntxx1/t-1=(2lnt-t)/t`

Hence `(df)/(dx)=(sint+tcost)/((2lnt-t)/t)=(tsint+t^2cost)/(2lnt-t)`