Question

Let y be an implicit function of x defined by `x^{2x} - 2x^x coty - 1 = 0`. Then, y'(1) equals ______ 

Options

• -1

• 1

• log 2

• -log 2

Answer 1

Let y be an implicit function of x defined by `x^{2x} - 2x^x coty - 1 = 0`. Then, y'(1) equals -1.

Explanation:

`x^{2x} - 2x^x coty - 1 = 0` ...............(i)

Putting x = 1 in (i), we get

1 - 2cot y - 1 = 0 ⇒ cot y = 0 ⇒ `y = pi/2`

Differentiating (i) w.r.t. x, we get

`2x^{2x}(1 + logx) - 2x^x(1 + logx)coty + 2x^x cosec^2y . dy/dx = 0`

Putting x = 1 and y = `pi/2`, we get

`2 - 0 + 2dy/dx = 0 ⇒ dy/dx = -1`