Maths · Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions
Assertion STATEMENT 1: Let = \) , f^{\prime}(2)=-\frac{2}{5} \) Reason STATEMENT
Assertion STATEMENT 1: Let \( \boldsymbol{f}(\boldsymbol{x})= \) \( \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right), f^{\prime}(2)=-\frac{2}{5} \) Reason STATEMENT \( 2: \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)=\pi \) \( 2 \tan ^{-1} x \forall x>1 \)
- A. Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT
- B. Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT 1
- C. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE
- D. STATEMENT 1 is FALSE and STATEMENT 2 is TRUE
Step-by-step solution
Statement 2: For x>1, using the identity sin^{-1}(2x/(1+x^2)) = π - 2 tan^{-1} x. Statement 1: f(x) = sin^{-1}(2x/(1+x^2)), then for x>1, f(x)=π-2tan^{-1}x, so f'(x)=-2/(1+x^2). At x=2, f'(2)=-2/5, so both true and Statement 2 explains the derivative.