Question

Find the inverse of the function and differentiate the inverse in two ways. (i) Differentiate the inverse function directly. (ii) d/dx f ⁻¹ (x) = 1/ f' [f ⁻¹ (x)] to find the derivative of the inverse. f(x) = √7x - 2 , x≥ 2/7 The inverse of f(x) is f⁻¹ (x) = x²/7 + 2/7 x≥ 0 (i) To find d/dx (f ⁻¹ (x)) directly, use the power rule. (ii) To use d/dx (f ⁻¹ (x)) = 1/ f' [f⁻¹ (x)] t find the derivative, find f' (x) f' (x) = ___

Answer 1

The derivative of the function f(x) = √7x - 2 is found using the power rule for derivatives and chain rule. Since our function can be rewritten as (7x)^(1/2) - 2, applying the chain rule gives us that the derivative f'(x) = 1/2 * (7x)^(-1/2) * 7, which simplifies to (sqrt(7))/(2*sqrt(x)).

Next, let's find the derivative of the inverse function f⁻¹ (x) = x²/7 + 2/7 directly. The derivative of a sum of functions is the sum of their derivatives, and using the power rule for differentiation (for x^n, the derivative is n*x^(n-1)) yields the derivative of f⁻¹ (x) as (2x/7).

Lastly, we want to find the derivative of f⁻¹ (x) using the formula 1/ f' [f ⁻¹ (x)]. We substitute f ⁻¹ (x) into f'(x), yielding a final derivative 1 / f' [f ⁻¹ (x)] = 0.152720709664243*sqrt(7)*sqrt(0.0408163265306122*(0.5*x^2 + 1)^2 + 1).

So, we have found the values of the original function derivative, the inverse function derivative found directly, and the inverse function derivative found using the formula, which are (sqrt(7)/(2*sqrt(x)), 2*x/7, and 0.152720709664243*sqrt(7)*sqrt(0.0408163265306122*(0.5*x^2 + 1)^2 + 1), respectively.