(a) Find y^' by implicit differentiation. (b) Solve the...

(a) Find $y^{\prime}$ by implicit differentiation. (b) Solve the equation explicitly for $y$ and differentiate to get $y^{\prime}$ in terms of $x$. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for $y$ into your solution for part (a). $\frac{2}{x}-\frac{1}{y}=4$

(a) Find $y^{\prime}$ by implicit differentiation.
(b) Solve the equation explicitly for $y$ and differentiate to get $y^{\prime}$ in terms of $x$.
(c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for $y$ into your solution for part (a).
$\frac{2}{x}-\frac{1}{y}=4$

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used when a function is given in an implicit form, meaning that the dependent variable is not isolated on one side of the equation. In this method, both sides of the equation are differentiated with respect to the independent variable, applying the chain rule to terms involving the dependent variable. This results in an equation that includes both the derivative of the dependent variable and the variable itself, which can be solved for the derivative.

Solving for the Explicit Form

Solving for the explicit form involves rearranging an implicit equation to isolate the dependent variable as an explicit function of the independent variable. This explicit form allows for straightforward differentiation using standard rules of differentiation, as the function is now directly expressed in terms of the independent variable.

Differentiation of Explicit Functions

When a function is given explicitly, its derivative is calculated using standard rules such as the power rule, product rule, or chain rule. This process generally is more straightforward than implicit differentiation since the dependent variable is directly expressed in terms of the independent variable.

Consistency Check Through Substitution

A consistency check involves substituting the explicit solution back into the derivative obtained via implicit differentiation to verify that both methods yield the same result. This step ensures that the implicit differentiation was performed correctly and that the explicit solution is valid, thereby reinforcing the correctness of the differentiation process.

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