The derivative of a function $$f(x)$$ is defined as $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$. For $$f(x) = x^3$$, this becomes $$f'(x) = \lim_{h \to 0} \frac{(x+h)^3 - x^3}{h}$$.

The derivative notation $$P'(t)$$ represents the instantaneous rate of change of the population with respect to time. Therefore, $$P'(5) = 2.1$$ means that at $$t=5$$ (the year 2015), the population was increasing at a rate of 2.1 thousand people per year, which is 2,100 people per year.

The derivative of a function $$f(x)$$ is defined as $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$. For $$f(x) = 5x^2 - 3$$, $$f(x+h) = 5(x+h)^2-3$$. Substituting these into the definition gives the expression in choice A. Choice B incorrectly omits the constant term from the function evaluation.

The alternative definition of the derivative of a function $$f$$ at a point $$a$$ is $$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}$$. Comparing this to the given limit, we can identify $$f(x) = \cos(x)$$ and $$a=2$$. The limit represents $$f'(2)$$.

This question requires interpreting derivative notation using function notation. When y=f(x), the derivative at x=c represents the instantaneous rate of change at that point. The notation f'(c) correctly expresses this derivative using standard function notation with prime symbol. This represents how the function f changes instantaneously at the point x=c, giving both the slope of the tangent line and the instantaneous rate of change. Function notation with prime is one of the most common ways to express derivatives at specific points. Choice A represents the average rate of change between x=0 and x=c, which gives a different value than the instantaneous rate at x=c. When using function notation for derivatives, the prime symbol indicates differentiation and the argument indicates the evaluation point.

This question requires interpreting derivative notation for instantaneous rate of change of mass. When m(t) represents mass as a function of time, the instantaneous rate of change at t=a is the derivative of m with respect to t evaluated at that point. The notation m'(a) correctly represents this derivative using prime notation, showing how mass changes instantaneously at t=a. This could represent rates like dissolution, accumulation, or chemical reaction rates in various contexts. Choice A represents the average rate of change of mass from t=0 to t=a, which doesn't capture the instantaneous behavior at the specific time t=a. For instantaneous rates in scientific contexts, use derivative notation to express precise rates of change at specific moments.

This question involves interpreting derivative notation in the context of how area changes with respect to radius. When A(r) gives area as a function of radius, the rate at which area changes with respect to r at r=2 is the derivative A'(2). This represents the instantaneous rate of change of area per unit change in radius at that specific value. In geometric contexts, this could represent concepts like marginal area or sensitivity of area to radius changes. Choice B represents the average rate of change of area between r=1 and r=2, which doesn't capture the instantaneous behavior at r=2 specifically. For rates of change in geometric contexts, use derivative notation to express how one quantity changes instantaneously with respect to another.

This question requires interpreting derivative notation in the context of dy/dx at a specific point. When y=f(x) is differentiable, dy/dx represents the derivative of y with respect to x, and we need this evaluated at x=4. The notation $\left.\dfrac{dy}{dx}\right|_{x=4}$ correctly expresses this using Leibniz notation with the evaluation bar. This shows the derivative of the dependent variable y with respect to the independent variable x, evaluated at the specific point x=4. Choice A represents the average rate of change between x=3 and x=4, which approximates but doesn't equal the instantaneous rate. For Leibniz notation derivatives, use the evaluation bar to specify the point of interest.

This question involves interpreting Leibniz notation when y is defined as p(t). When y=p(t) is differentiable, the expression dy/dt at t=6 represents the derivative of y with respect to t evaluated at that point. The notation $\left.\dfrac{dy}{dt}\right|_{t=6}$ correctly expresses this using Leibniz notation with evaluation. This shows the instantaneous rate of change of the dependent variable y with respect to the independent variable t at the specific time t=6. Choice A represents the average rate of change between t=0 and t=6, which doesn't equal the instantaneous rate at t=6. When using Leibniz notation with function definitions like y=p(t), maintain consistent variable notation throughout.

This question involves interpreting derivative notation in the context of speed. When x represents time and y represents distance with y=d(x), the instantaneous speed at x=6 is the derivative of distance with respect to time at that point. The notation $\left.\dfrac{dy}{dx}\right|_{x=6}$ correctly represents this using Leibniz notation with evaluation. Since speed is the magnitude of velocity, and velocity is the derivative of position (distance) with respect to time, this notation captures the instantaneous speed. Choice A represents the average speed between x=5 and x=6, which approximates but doesn't equal the instantaneous speed at x=6. For instantaneous physical quantities like speed, use derivative notation with proper evaluation symbols.