Question

How do you differentiate the following parametric equation: `(3t-sin(t/3-pi/4), 2tcos(pi/3-t/4))`?

Answer 1

`(dy)/(dx)=(3(4cos(pi/3-t/4)-tsin(pi/3-t/4)))/(18-2cos(t/3-pi/4))`

Explanation:

In a parametric equation of the type `f(x(t),y((t))`, `(dy)/(dx)=((dy)/(dt))/((dx)/(dt))`.

Here `x(t)=3t-sin(t/3-pi/4)` and hence `(dx)/(dt)=3-cos(t/3-pi/4)xx1/3`

and as `y(t)=2tcos(pi/3-t/4)` `(dy)/(dt)=2(cos(pi/3-t/4)-tsin(pi/3-t/4)xx(-1/4))`

Hence, `(dy)/(dx)=(2(cos(pi/3-t/4)+t/4sin(pi/3-t/4)))/(3-cos(t/3-pi/4)xx1/3)`

= `(2/4(4cos(pi/3-t/4)+tsin(pi/3-t/4)))/(1/3(9-cos(t/3-pi/4)))`

= `(3(4cos(pi/3-t/4)-tsin(pi/3-t/4)))/(18-2cos(t/3-pi/4))`