Evaluate and simplify y'.
y = (3t²−1 / 3t²+1)^−3

Verified step by step guidance

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.
2. 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}.
3. Step 3: Differentiate the outer function with respect to u. The derivative of u^{-3} with respect to u is -3u^{-4}.
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.
5. 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.

Key Concepts

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.

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.

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.