Implicit Differentiation
- Page ID
- 217636
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Implicit Differentiation Implicit and Explicit functions An explicit function is an function expressed as \( y = f(x)\) such as
To find \( \frac{dy}{dx} \), we proceed as follows:
Example Solution
Example: Solution:
Exercises:
Application
Example Suppose that the demand function for a boat shop is given by \( p = -0.01x^3 + x + 10,000 \) Find the rate of change of \( x \) with respect to p when \( x = 20 \). A boat craftsman can think of this question as who fast will the number of boats she will need to build change as the price is increased. Solving for \( x \) in terms of \( p \) is nearly impossible. Instead, we can differentiate implicitly. \( 1 = -0.03x^2 x' + x' \) Now plug in 20 for \( x \) to get \( 1 = -0.03(20)^2 x' + x' \) \( = -12x' + x' = -11x' \) \( x' = \frac{-1}{11} \) The rate of change is \( \frac{-1}{11} \) boats per dollar increase.
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