1. Differentiate x^sinx w.r.t.x 2. If x = at^2, y = 2at, then find dy/dx

Question

1. Differentiate x^sinx w.r.t.x 
2. If x = at², y = 2at, then find \(\frac{dy}{dx}\) 
3. If y = sin^-1(cosx) + cos^-1(sinx), then find \(\frac{dy}{dx}\).

Answer 1

1. Let y = x^sinx, take log on both sides,
log y = sinx logx differentiate w. r.t.x, we get

2.

3. Given, 

y = sin^-1(cosx) + cos^-1(sinx)

Differentiate w.r.t. x, we get \(\frac{dy}{dx}\) = -2