Question

If x = sin θ, y = tan θ, then find `("d"y)/("d"x)`.

Answer 1

x = sin θ

Differentiating w. r. t. θ, we get

`("d"x)/("d"theta) = "d"/("d"theta) (sintheta)` = cos θ

y = tan θ

Differentiating w. r. t. θ, we get

`("d"y)/("d"theta) = "d"/("d"theta) (tantheta)`  = sec² θ

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

= `(sec^2theta)/(cos theta)`

= sec³ θ