If u = sin^-1((2x)/(1 + x^2)) and v = tan^-1((2x)/(1 + x^2)), where -1 < x < 1, then write the value of du/dv.

Question

If u = sin^-1\(\Big(\cfrac{2\text x}{1+\text x^2}\Big)\) and v = tan^-1\(\Big(\cfrac{2\text x}{1+\text x^2}\Big)\), where -1 < x < 1, then write the value of \(\cfrac{du}{d\mathrm x} \).

Answer 1

We know,

Using the chain rule of differentiation,

Using Chain Rule of Differentiation,

Dividing numerator and denominator by (1+x²)²,

=sec u (1+tan u )