If y = ⁻¹ ( 25 - x^2 25 + x^2 ) , then y'(1) is...

Step-by-step solution

Given y = ⁻¹ ( 25 - x^2 25 + x^2 ) Substitute x = 5 , which gives = ⁻¹ ( x 5 ) y = ⁻¹ ( 25 - 25 ^2 25 + 25 ^2 ) y = ⁻¹ ( 1 - ^2 1 + ^2 ) y = ⁻¹( 2 ) y = ⁻¹ ( ( 2 - 2 ) ) For x = 1 , = ⁻¹ ( 1 5 ) , so 2 (0, 2 ) , which lies in the principal domain of ⁻¹ . y = 2 - 2 = 2 - 2 ⁻¹ ( x 5 ) Differentiating with respect to x : dy dx = -2 1 1 + ( x 5 )^2 1 5 = - 10 25 + x^2 Substituting x = 1 : y'(1) = - 10 25 + 1 = - 10 26 = - 5 13 Answer: - 5 13