Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.2: Derivatives Of Products And Quotients
Problem 37E
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Question
![**Exercise: Finding the Derivative of the Inverse Function**
**Problem Statement:**
Calculate \( g'(x) \), where \( g(x) \) is the inverse of \( f(x) = \frac{x}{x+2} \).
**Solution:**
To find \(g'(x)\), follow these steps:
1. **Understand the Relationship Between \(f(x)\) and \(g(x)\):**
Since \(g(x)\) is the inverse of \(f(x)\), we have \(f(g(x)) = x\) and \(g(f(x)) = x\).
2. **Implicit Differentiation:**
Differentiate both sides of \(f(g(x)) = x\) with respect to \(x\):
\[
f'(g(x)) \cdot g'(x) = 1.
\]
3. **Solve for \(g'(x)\):**
\[
g'(x) = \frac{1}{f'(g(x))}.
\]
4. **Find \(f'(x)\):**
Calculate the derivative of \(f(x)\) with respect to \(x\):
\[
f(x) = \frac{x}{x+2}.
\]
Using the quotient rule for differentiation, we get:
\[
f'(x) = \frac{(x+2) - x}{(x+2)^2} = \frac{2}{(x+2)^2}.
\]
5. **Substitute Back:**
\(
f(g(x)) = x \Rightarrow g(x) = f^{-1}(x).
\)
Thus,
\[
g'(x) = \frac{(g(x)+2)^2}{2}.
\]
Plugging the inverse function \(g(x)\) back can often be complex, but the critical part is calculating and using \(f'\) correctly as shown.
**Final Answer Box:**
\[
g'(x) = \boxed{\frac{(g(x)+2)^2}{2}}.
\]
The right-hand side empty box with a pencil icon signifies that the final value of \(g'(x)\) needs to be calculated from the derived expression using the correct inverse function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1b16e79-59da-42ec-8251-0c12184e0a83%2F42f9d0f3-a2a0-4158-aeda-0baf924b912e%2F6gm5p8u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise: Finding the Derivative of the Inverse Function**
**Problem Statement:**
Calculate \( g'(x) \), where \( g(x) \) is the inverse of \( f(x) = \frac{x}{x+2} \).
**Solution:**
To find \(g'(x)\), follow these steps:
1. **Understand the Relationship Between \(f(x)\) and \(g(x)\):**
Since \(g(x)\) is the inverse of \(f(x)\), we have \(f(g(x)) = x\) and \(g(f(x)) = x\).
2. **Implicit Differentiation:**
Differentiate both sides of \(f(g(x)) = x\) with respect to \(x\):
\[
f'(g(x)) \cdot g'(x) = 1.
\]
3. **Solve for \(g'(x)\):**
\[
g'(x) = \frac{1}{f'(g(x))}.
\]
4. **Find \(f'(x)\):**
Calculate the derivative of \(f(x)\) with respect to \(x\):
\[
f(x) = \frac{x}{x+2}.
\]
Using the quotient rule for differentiation, we get:
\[
f'(x) = \frac{(x+2) - x}{(x+2)^2} = \frac{2}{(x+2)^2}.
\]
5. **Substitute Back:**
\(
f(g(x)) = x \Rightarrow g(x) = f^{-1}(x).
\)
Thus,
\[
g'(x) = \frac{(g(x)+2)^2}{2}.
\]
Plugging the inverse function \(g(x)\) back can often be complex, but the critical part is calculating and using \(f'\) correctly as shown.
**Final Answer Box:**
\[
g'(x) = \boxed{\frac{(g(x)+2)^2}{2}}.
\]
The right-hand side empty box with a pencil icon signifies that the final value of \(g'(x)\) needs to be calculated from the derived expression using the correct inverse function.
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