Ask AI: Differentiate with respect to x. a. y=x^2sin^-1(x/2) b. y=tan^-1(tanx)

To differentiate the given functions with respect to x, we will apply the chain rule and other differentiation rules as needed. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. a. y = x^2*sin^(-1)(x/2) Let's differentiate step by step: 1. Differentiate x^2 with respect to x: d(x^2)/dx = 2x. 2. Differentiate sin^(-1)(x/2) with respect to (x/2): d(sin^(-1)(x/2))/d(x/2) = 1/√(1-(x/2)^2). 3. Differentiate (x/2) with respect to x: d(x/2)/dx = 1/2. Now we'll use the chain rule: dy/dx = (2x) * (1/√(1-(x/2)^2)) * (1/2). Combining, we get: dy/dx = x/√(1-(x^2)/4). b. y = tan^(-1)(tan(x)) Since tan^(-1)(tan(x)) simplifies to x, within their respective domains: dy/dx = d(x)/dx = 1.