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