Question

If x = a sec³θ and y = a tan³θ

(i) Find \(\frac{dx}{dθ}, \frac{dy}{dθ}\)

(ii) Find \(\frac{dy}{dx}\)

(iii) Find \(\frac{d^2y}{dx^2}\)

(iv) Show that (\(\frac{dy}{dx}\))_{\(​​θ = \frac{π}{4} \)} = \(\frac{1}{12a}\)

Answer 1

 (i) Given, 

x = a sec³θ

Differentiate w.r.to θ, we get;

\(\frac{dx}{dθ}\) = 3a sec²θ. secθ. tanθ = 3a sec³ θ. tan θ

Given, 

y = a tan³θ .

Differentiating w.r.to θ, we get

\(\frac{dx}{dθ}\) = 3a tan² θ. sec²θ.

(iii) We have, 

\(\frac{dy}{dx}\) = sinθ

Differentiating w.r.to x, we get

(iv) We have,