Question

If x = a sec³θ and y = a tan³θ, find `("d"y)/("d"x)` at θ = `pi/3`

Answer 1

We have x =  sec³θ and y = a tan³θ

Differentiating w.r.t. θ , we get

`("d"x)/("d"theta) = 3"a" sec^2 theta  "d"/("d"theta) (sec theta)`

= 3a sec³θ tanθ

And `("d"y)/("d"theta) = 3"a" tan^2 theta "d"/("d"theta) (tan theta)`

= 3a tan³θ sec²θ.

Thus `("d"y)/("d"x) = (("d"y)/("d"theta))/(("d"x)/("d"theta))`

= `tantheta/sectheta`

= sin θ

Hence, `(("d"y)/("d"x))_("at" theta = pi/3) = sin  pi/3 = sqrt(3)/2`.