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 Integrals4h 44m
- 9. Graphical Applications of Integrals2h 27m
- 10. Physics Applications of Integrals 2h 22m
3. Techniques of Differentiation
The Chain Rule
Problem 36
Textbook Question
Evaluate and simplify y'.
y = (3t²−1 / 3t²+1)^−3

1
Step 1: Recognize that the function y = \left(\frac{3t^2 - 1}{3t^2 + 1}\right)^{-3} is a composition of functions, specifically a power function and a rational function. To find y', we will use the chain rule.
Step 2: Apply the chain rule. The chain rule states that if y = f(g(t)), then y' = f'(g(t)) \cdot g'(t). Here, let u = \frac{3t^2 - 1}{3t^2 + 1}, so y = u^{-3}.
Step 3: Differentiate the outer function with respect to u. The derivative of u^{-3} with respect to u is -3u^{-4}.
Step 4: Differentiate the inner function u = \frac{3t^2 - 1}{3t^2 + 1} with respect to t using the quotient rule. The quotient rule states that if u = \frac{v}{w}, then u' = \frac{v'w - vw'}{w^2}. Here, v = 3t^2 - 1 and w = 3t^2 + 1.
Step 5: Combine the results from Steps 3 and 4. Substitute the derivative of the inner function into the chain rule expression from Step 2 to find y'. Simplify the expression if possible.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve at any given point. In this context, we need to apply the rules of differentiation to find y', the derivative of the given function y.
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Chain Rule
The chain rule is a formula for computing the derivative of a composite function. It states that if a function y can be expressed as a composition of two functions, say u(t) and v(u), then the derivative of y with respect to t is the product of the derivative of y with respect to u and the derivative of u with respect to t. This rule is essential for differentiating functions raised to a power, as seen in the given expression.
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Quotient Rule
The quotient rule is used to differentiate functions that are expressed as the ratio of two other functions. It states that if y = f(t)/g(t), then the derivative y' is given by (f'g - fg')/g². In this problem, since y is a fraction, applying the quotient rule will be necessary to correctly differentiate the numerator and denominator before simplifying the result.
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