21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(t) = 3t⁴; a= -2, 2

Verified step by step guidance

1. Step 1: Recall the definition of the derivative using limits. The derivative of a function f at a point t is given by the limit: f'(t) = \(\lim\)_{h \(\to\) 0} \(\frac{f(t+h) - f(t)}{h}\).
2. Step 2: Substitute the given function f(t) = 3t^4 into the derivative definition. This gives: f'(t) = \(\lim\)_{h \(\to\) 0} \(\frac{3(t+h)^4 - 3t^4}{h}\).
3. Step 3: Expand the expression (t+h)^4 using the binomial theorem: (t+h)^4 = t^4 + 4t^3h + 6t^2h^2 + 4th^3 + h^4.
4. Step 4: Substitute the expanded form back into the limit expression: f'(t) = \(\lim\)_{h \(\to\) 0} \(\frac{3(t^4 + 4t^3h + 6t^2h^2 + 4th^3 + h^4) - 3t^4}{h}\).
5. Step 5: Simplify the expression by canceling terms and factoring out h from the numerator, then evaluate the limit as h approaches 0 to find f'(t).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In mathematical terms, the derivative f'(t) is given by the limit: f'(t) = lim(h→0) [f(t+h) - f(t)] / h.

Limit

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It is essential for defining derivatives and integrals. The limit helps in understanding the function's behavior at points where it may not be explicitly defined, allowing for the calculation of derivatives using the definition involving the difference quotient.

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 the power rule, the derivative f'(t) is calculated as f'(t) = n * t^(n-1). This rule simplifies the process of finding derivatives for polynomial functions, making it easier to compute derivatives quickly.