Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation3h 18m
- 4. Applications of Derivatives2h 38m
- 5. Graphical Applications of Derivatives6h 2m
- 6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
- 7. Antiderivatives & Indefinite Integrals1h 26m
3. Techniques of Differentiation
The Chain Rule
Problem 84e
Textbook Question
Use the given graphs of f and g to find each derivative. <IMAGE>
d/dx (g(f(x))) |x=1
![](/channels/images/assetPage/verifiedSolution.png)
1
Step 1: Recognize that you need to find the derivative of a composite function, g(f(x)), at x = 1. This requires the use of the chain rule.
Step 2: The chain rule states that the derivative of g(f(x)) with respect to x is g'(f(x)) * f'(x).
Step 3: Evaluate f(x) at x = 1 using the graph of f to find f(1).
Step 4: Use the graph of g to find g'(f(1)), which is the derivative of g at the point f(1).
Step 5: Use the graph of f to find f'(1), which is the derivative of f at x = 1. Multiply g'(f(1)) by f'(1) to find the derivative of g(f(x)) at x = 1.
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