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Chapter 2: Problem 66

In your own words, state the guidelines for implicit differentiation.

Short Answer

Expert verified
Implicit differentiation entails: 1) Choosing to use this method when it is hard or impossible to solve an equation for a particular variable before differentiation. 2) Applying the Chain Rule by multiplying a function of 'y' by \( y' \) or \(\frac{dy}{dx}\) when differentiated. 3) Differentiating with respect to 'x', thinking of 'y' as a function of 'x'. 4) After differentiation, attempting to solve for \( y' \) or \(\frac{dy}{dx}\) to get the derivative.

Step by step solution

01

Guideline 1: When to use implicit differentiation

Implicit differentiation should be used when it is impractical or impossible to explicitly solve an equation for a certain variable before differentiating.
02

Guideline 2: Applying the Chain Rule

During the process of implicit differentiation, the Chain Rule is applied. This means that whenever a function of 'y' is differentiated, one must also multiply by \( y' \) or \(\frac{dy}{dx}\). This is because 'y' is considered to be a function of 'x', even if that function is unspecified.
03

Guideline 3: Differentiating with respect to 'x'

In implicit differentiation, all differentiation takes place with respect to the variable 'x'. Even if a function includes terms involving 'y', those terms are differentiated as if 'y' is a function of 'x', not a variable on its own.
04

Guideline 4: Solving for 'y'

After differentiation, the resulting equation should be solved for \( y' \) or \(\frac{dy}{dx}\) to get the derivative of y with respect to x, if it is possible.

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