For n N if y =a x^ n +1 + b x^ - n , then x^2 ~d ^2 y d x^2 =

Step-by-step solution

The first derivative of y = a x^ n+1 + b x^ -n is found using the power rule: dy dx = a(n+1)x^ n - b n x^ -n-1 Differentiating again yields the second derivative: d^2y dx^2 = a n(n+1)x^ n-1 + b n(n+1)x^ -n-2 Multiplying by x^2 gives: x^2 d^2y dx^2 = a n(n+1)x^ n+1 + b n(n+1)x^ -n = n(n+1)(a x^ n+1 + b x^ -n ) Recognizing the original function y in the expanded form: x^2 d^2y dx^2 = n(n+1)y This confirms that the correct relationship is option C .