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
- 8. Definite Integrals3h 25m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Inverse Trigonometric Functions
Problem 3.R.43
Textbook Question
9–61. Evaluate and simplify y'.
y = x²+2x tan^−1(cot x)
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1
Identify the function y = x² + 2x tan^−1(cot x) and differentiate it with respect to x to find y'.
Use the product rule for differentiation on the term 2x tan^−1(cot x) since it is a product of two functions: 2x and tan^−1(cot x).
For the derivative of tan^−1(cot x), apply the chain rule, remembering that cot x = cos x/sin x and its derivative is -csc²x.
Combine the results from the differentiation of both parts to express y' in terms of x.
Substitute y' back into the equation y' * y to evaluate and simplify the expression.
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