This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
ScanSolve AI Answer
You're on a roll — 6. Differentiate f(x) = (^-1 ( x)). Step 1: Apply the chain rule. The outermost function is (u), where u = ^-1( x). The derivative of (u) is (1)/(u) (du)/(dx). f'(x) = (1)/(^-1)( x) · (d)/(dx)(^-1( x)) Step 2: Differentiate ^-1(v), where v = x. The derivative of ^-1(v) is (1)/(sqrt(1+v^2)) (dv)/(dx). (d)/(dx)(^-1( x)) = (1)/(sqrt(1+( x)^2)) · (d)/(dx)( x) Step 3: Differentiate x. The derivative of x is ^2 x. (d)/(dx)( x) = ^2 x Step 4: Substitute the derivatives back into the expression for f'(x). f'(x) = (1)/(^-1)( x) · (1)/(sqrt(1+^2 x)) · ^2 x Step 5: Simplify the expression using the trigonometric identity 1+^2 x = ^2 x. f'(x) = (1)/(^-1)( x) · (1)/(sqrt(^2 x)) · ^2 x Assuming x > 0, then sqrt(^2 x) = x. f'(x) = (1)/(^-1)( x) · (1)/( x) · ^2 x f'(x) = ( x)/(^-1)( x) The final answer is ( x)/(^-1)( x). What's next?