If y= ⁻¹ ( 3-2 x 1+6 x ) then d y d x is

Step-by-step solution

The given expression is y = ⁻¹ ( 3-2x 1+6x ) . We can rewrite the argument of the inverse tangent function as follows: 3-2x 1+6x = 3-2x 1+(3)(2x) Using the identity ⁻¹ ( a-b 1+ab ) = ⁻¹(a) - ⁻¹(b) , we set a=3 and b=2x . Thus, y = ⁻¹(3) - ⁻¹(2x) . Differentiating both sides with respect to x : dy dx = d dx ( ⁻¹(3)) - d dx ( ⁻¹(2x)) Since ⁻¹(3) is a constant, its derivative is 0 . Using the chain rule for ⁻¹(2x) , we get: dy dx = 0 - 1 1+(2x)^2 d dx (2x) dy dx = - 1 1+4x^2 2 dy dx = - 2 1+4x^2 Answer: - 2 1+4 x^2