Final answer:
To find y' by implicit differentiation, differentiate both sides of the equation. Solve the equation explicitly for y and differentiate it to find y' in terms of x. Check the solutions using substitution.
Step-by-step explanation:
To find y' by implicit differentiation, we differentiate both sides of the equation concerning x. However, since y is a function of x, we need to use the chain rule. The derivative of 4/x concerning x is -4/x^2, and the derivative of 1/y concerning x is -y'/y^2. Therefore, the equation becomes -4/x^2 - y'/y^2 = 0. Rearranging, we get y' = (-4/x^2) * y^2.
To solve the equation explicitly for y, we rearrange the equation 4/x - 1/y = 5 and solve for y in terms of x. Multiply both sides by xy to eliminate the denominators, and then re-arrange the resulting quadratic equation to get y= (-5x ± √(25x^2 - 64x))/8x. To differentiate this equation and get y' in terms of x, we can use the quotient rule. Finally, we can check the solutions to parts (a) and (b) by substituting the expression for y from part (b) into the equation from part (a) and verifying that it satisfies the equation.