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).