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
Product and Quotient Rules
Problem 66a
Textbook Question
Population growth Consider the following population functions.
a. Find the instantaneous growth rate of the population, for t≥0.
p(t) = 600 (t²+3/t²+9)
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1
Step 1: Identify the function for which you need to find the instantaneous growth rate. The function given is \( p(t) = 600 \left( \frac{t^2 + 3}{t^2 + 9} \right) \).
Step 2: To find the instantaneous growth rate, you need to compute the derivative of the function \( p(t) \) with respect to \( t \). This involves using the quotient rule for differentiation.
Step 3: Recall the quotient rule: if you have a function \( \frac{u(t)}{v(t)} \), its derivative is \( \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} \). Here, \( u(t) = t^2 + 3 \) and \( v(t) = t^2 + 9 \).
Step 4: Differentiate \( u(t) \) and \( v(t) \) separately. \( u'(t) = 2t \) and \( v'(t) = 2t \).
Step 5: Apply the quotient rule: \( p'(t) = 600 \left( \frac{(2t)(t^2 + 9) - (t^2 + 3)(2t)}{(t^2 + 9)^2} \right) \). Simplify the expression to find the derivative.
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