Question

Find the chain rule derivative of sin^(-1)(f(x)).

Answer 1

The derivative of sin^(-1)(f(x)) using the chain rule is (f'(x))/(√(1 - (f(x))^2)). This result is obtained by differentiating the outer function with respect to its inner function and then multiplying by the derivative of the inner function.

Step-by-step explanation:

The question asks to find the chain rule derivative of sin-1(f(x)). To apply the chain rule, we start by differentiating the outer function, which is sin-1(u), with respect to its argument u, and then we multiply that by the derivative of the inner function, f(x), with respect to x. In more mathematical terms:

If y = sin-1(u) and u = f(x), then by the chain rule:

• dy/du = 1/√(1 - u2)
• du/dx = f'(x)
• Therefore, dy/dx = dy/du * du/dx = (1/√(1 - u2)) * f'(x) = (1/√(1 - (f(x))2)) * f'(x)

So, the derivative of sin-1(f(x)) with respect to x using the chain rule is (f'(x))/(√(1 - (f(x))2)).