Let u = sin-1\((2\text x\sqrt{1-\text x^2})\) and v = sec-1\(\Big(\cfrac{1}{\sqrt{1-\text x^2}}\Big).\)
We need to differentiate u with respect to v that is find \(\cfrac{du}{dv}
\).
We have u = sin-1\((2\text x\sqrt{1-\text x^2})\)
By substituting x = sin θ, we have
⇒ u = sin-1(2 sin θ cos θ)
⇒ u = sin-1(sin 2θ)
Now, we have v = sec-1\(\Big(\cfrac{1}{\sqrt{1-\text x^2}}\Big)\)
By substituting x = sin θ, we have
Given x ∈ \(\Big(0,\cfrac{1}{\sqrt2}\Big)\)
However, x = sin θ
Hence, u = sin–1(sin 2θ) = 2θ.
⇒ u = 2sin–1(x)
On differentiating u with respect to x, we get
On differentiating v with respect to x, we get
We have
Thus,
\(\cfrac{du}{dv} = 2
\)
