21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(s) = 4s³+3s; a= -3, -1

Verified step by step guidance

1. Step 1: Recall the definition of the derivative using limits. The derivative of a function f(s) at a point s is given by the limit: f'(s) = \(\lim\)_{h \(\to\) 0} \(\frac{f(s+h) - f(s)}{h}\).
2. Step 2: Substitute the given function f(s) = 4s^3 + 3s into the limit definition. This gives us: f'(s) = \(\lim\)_{h \(\to\) 0} \(\frac{(4(s+h)^3 + 3(s+h)) - (4s^3 + 3s)}{h}\).
3. Step 3: Expand the expression (s+h)^3 using the binomial theorem: (s+h)^3 = s^3 + 3s^2h + 3sh^2 + h^3. Substitute this into the limit expression.
4. Step 4: Simplify the expression by distributing and combining like terms. This involves expanding 4(s+h)^3 and 3(s+h), then subtracting 4s^3 + 3s.
5. Step 5: Factor out h from the numerator and then cancel it with the h in the denominator. Finally, take the limit as h approaches 0 to find the derivative f'(s).

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. The derivative is often denoted as f'(s) and represents the slope of the tangent line to the function's graph at a given point.

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 used to define derivatives and integrals. In the context of finding a derivative, the limit helps to determine the instantaneous rate of change of the function at a specific point by examining the function's values as they get arbitrarily close to that point.

Power Rule

The Power Rule is a basic differentiation rule that simplifies the process of finding the derivative of polynomial functions. It states that if f(s) = s^n, where n is a real number, then the derivative f'(s) = n * s^(n-1). This rule is particularly useful for functions like f(s) = 4s³ + 3s, allowing for quick calculation of derivatives without using the limit definition directly.