MHT CET202520 Apr 2025Evening ShiftMathematicsDifferentiationActual
If u = ⁻¹ x ⁻¹ x+1 and v = ⁻¹ ( ⁻¹ x ) then du dv =
Options
- A1
- B1+ ( ⁻¹ x )^2 (1+ ⁻¹ x )^2
- C⁻¹ x (1+ ⁻¹ x )^2
- D1 (1+ ⁻¹ x )^2
Correct answer
B. 1+ ( ⁻¹ x )^2 (1+ ⁻¹ x )^2
Step-by-step solution
Let y = ⁻¹ x , so we can rewrite the functions as: u = y y+1 v = ⁻¹ y The derivative du dv can be found using the chain rule: du dv = du/dy dv/dy Differentiating u with respect to y using the quotient rule: du dy = (1)(y+1) - y(1) (y+1)^2 = 1 (y+1)^2 Differentiating the inverse tangent function: dv dy = 1 1+y^2 Substituting these derivatives into the chain rule expression: du dv = 1/(y+1)^2 1/(1+y^2) = 1+y^2 (y+1)^2 Replacing y with ⁻¹ x gives the final result: