Consider the following.
\[
4 x^{5}+y^{3}=9 x
\]
(a) Find $y^{\prime}$ by implicit differentiation.
\[
y^{\prime}=
\]
(b) Solve the equation explicitly for $y$ and differentiate to get $y^{\prime}$ in terms of $x$.
\[
y^{\prime}=
\]
(c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for $y$ into your solution for part (a).
\[
y^{\prime}=
\]
Answer
\(\boxed{y' = \frac{9 - 20x^4}{3y^2}}\) is the final answer.
Step 1 ：Differentiate both sides of the equation \(4x^5 + y^3 = 9x\) with respect to x. The derivative of a sum of terms is the sum of the derivatives of each term. The derivative of a constant times a function is the constant times the derivative of the function. The derivative of x to the power of n is n times x to the power of n-1. The derivative of y with respect to x is denoted as y'.
Step 2 ：Applying these rules, we get \(20x^4 + 3y^2y' = 9\).
Step 3 ：Solving this equation for y', we get \(y' = \frac{9 - 20x^4}{3y^2}\).
Step 4 ：\(\boxed{y' = \frac{9 - 20x^4}{3y^2}}\) is the final answer.
