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
5:19 minutes
Problem 23
Textbook Question
Textbook Question5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = tan 5x²
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Composite Functions
A composite function is formed when one function is applied to the result of another function. In the context of the question, we express the function y = tan(5x²) as y = f(g(x)), where g(x) is the inner function and f(u) is the outer function. Understanding how to identify these functions is crucial for applying the chain rule in differentiation.
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Chain Rule
The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if y = f(g(x)), then the derivative dy/dx can be calculated as dy/dx = f'(g(x)) * g'(x). This rule allows us to find the derivative of complex functions by breaking them down into simpler parts, which is essential for solving the given problem.
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Inner and Outer Functions
In a composite function, the inner function is the one that is applied first, while the outer function is applied to the result of the inner function. For the function y = tan(5x²), the inner function can be identified as g(x) = 5x² and the outer function as f(u) = tan(u). Recognizing these functions is key to correctly applying the chain rule and finding the derivative.
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