21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(t) = 3t⁴; a= -2, 2

Verified step by step guidance

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.
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).
3. 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.
4. Step 4: Substitute the given values of a into the derivative f'(t) to find f'(-2) and f'(2).
5. 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.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at a given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.

Power Rule

The power rule is a basic differentiation rule used to find the derivative of functions of the form f(t) = t^n, where n is a real number. According to this rule, the derivative f'(t) is given by n*t^(n-1). This rule simplifies the process of differentiation, especially for polynomial functions, making it easier to evaluate derivatives at specific points.

Evaluating Derivatives at Specific Points

Evaluating the derivative at specific points involves substituting the given values into the derivative function. For instance, once the derivative f'(t) is calculated, substituting a = -2 or a = 2 into f'(t) yields the slope of the tangent line at those points on the original function. This process is crucial for understanding the behavior of the function at specific locations.