If y =a^x ~b ^ 2 x-1 , then d ^2 y d x^2 is equal to

Step-by-step solution

To differentiate y = a^x b^ 2x-1 , simplify the expression by factoring constants: y = a^x b^ 2x b⁻¹ = (a b^2)^x b⁻¹ Differentiating using d dx (c^x) = c^x c : dy dx = b⁻¹ (a b^2)^x (a b^2) = y (a b^2) For the second derivative, differentiate dy dx while noting (a b^2) is constant: d^2y dx^2 = (a b^2) dy dx = y ( (a b^2))^2 This corresponds to option C.