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
Basic Rules of Differentiation
Problem 3.2.30b
Textbook Question
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(t) = 3t⁴; a= -2, 2
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1
Step 1: Identify the function f(t) = 3t^4 and the values of a for which we need to evaluate the derivative, which are a = -2 and a = 2.
Step 2: Find the derivative of the function f(t) with respect to t. Use the power rule for differentiation, which states that if f(t) = t^n, then f'(t) = n*t^(n-1).
Step 3: Apply the power rule to f(t) = 3t^4. The derivative f'(t) is found by multiplying the exponent by the coefficient and reducing the exponent by one.
Step 4: Substitute the given values of a into the derivative f'(t) to find f'(-2) and f'(2).
Step 5: Simplify the expressions obtained from substituting a = -2 and a = 2 into f'(t) to find the values of the derivative at these points.
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