The notation $$\frac{dx}{dy}$$ represents the derivative of $$x$$ with respect to $$y$$, which is the reciprocal of the derivative of $$y$$ with respect to $$x$$. The other three options, $$f'(x)$$, $$\frac{dy}{dx}$$, and $$y'$$, are all standard notations for the derivative of $$y$$ with respect to $$x$$.

The Leibniz notation $$\frac{dV}{dr}$$ represents the derivative of the volume function $$V$$ with respect to the radius variable $$r$$. A derivative represents an instantaneous rate of change. Therefore, $$\frac{dV}{dr}$$ is the instantaneous rate of change of the volume with respect to the radius. Option C represents $$\frac{dr}{dV}$$.

For any function or curve, the notation $$\frac{dy}{dx}$$ represents the derivative of $$y$$ with respect to $$x$$. Geometrically, this value gives the slope of the line tangent to the curve at a given point $$(x,y)$$. Option D describes $$\frac{dx}{dy}$$.

The expression in option B is the alternate form of the limit definition of the derivative at a point $$a$$. Option A is the difference quotient before the limit is taken. Option C is related to the definition of continuity, not the derivative. Option D represents the average rate of change over the interval $$[a, b]$$.

Recognizing and equating different forms of derivative notation is a key skill in calculus. The notation $(\left.\d$\frac{dy}{dx}$\right|{x=a}$) is the first derivative of y with respect to x evaluated at x=a. Given y=q(x), this is the same as q'(a) in prime notation, both signifying the derivative value at that point. Second derivatives would be denoted by q''(a) or d²y/dx², differentiating them clearly. A tempting distractor could be $(\left.\d$\frac{d^2y$$}{dx^$2$}$\right|{x=a}$), but it captures the second derivative, failing to equate to the first derivative shown. Always match the order of the derivative and the evaluation point when translating between notations.

Recognizing and equating different forms of derivative notation is a key skill in calculus. The notation $(\d$\frac{d^2y$$}{dx^2$}$) evaluated at x=1 represents the second derivative of y with respect to x at that point. Since y=g(x), this is equivalent to g''(1) in prime notation, both indicating the rate of change of the slope. First derivatives would use a single prime or dy/dx, distinguishing them from higher orders. A tempting distractor might be $(\left.\d$\frac{dy}{dx}$\right|_{x=1}$), but it only captures the first derivative, not the second, so it fails to match. Always match the order of the derivative and the evaluation point when translating between notations.

This problem requires converting from prime notation to Leibniz notation. The notation $R'(10)$ means "the derivative of revenue R evaluated at q = 10." Since R is a function of q, the derivative is $\frac{dR}{dq}$, and evaluating at q = 10 gives us $\left.\frac{dR}{dq}\right|{q=10}$, which is choice B. Choice C showing $\left.\frac{dq}{dR}\right|{q=10}$ has the variables in the wrong positions—this would be the reciprocal of the derivative we want. Remember that in Leibniz notation, the dependent variable (R) goes in the numerator and the independent variable (q) goes in the denominator.

This question tests your ability to recognize equivalent derivative notations. The notation $\frac{ds}{dt}\bigg|{t=3}$ means "the derivative of s with respect to t, evaluated at t = 3." This is exactly what choice C shows: $\left.\frac{ds}{dt}\right|{t=3}$ uses the vertical bar notation to indicate evaluation at t = 3. Choice E might seem tempting because it shows $\frac{ds}{dt}(t=3)$, but this notation is ambiguous—it could mean the derivative function multiplied by (t=3) rather than evaluation at that point. When converting between derivative notations, remember that the vertical bar clearly indicates "evaluate at" while parentheses can be ambiguous without proper context.

This question asks you to identify the prime notation equivalent of a Leibniz derivative. The expression $\left.\frac{dx}{dt}\right|{t=7}$ represents the derivative of x with respect to t, evaluated at t = 7. In prime notation, this is written as $x'(7)$, which is choice C. Choice B showing $\left.\frac{dt}{dx}\right|{t=7}$ would be the reciprocal of our derivative, representing how t changes with respect to x instead. To convert correctly between notations, remember that x'(a) always means the derivative of x evaluated at the independent variable equal to a.

This question requires recognizing that different derivative notations represent the same mathematical concept. The notation $f'(2)$ means "the derivative of function f evaluated at x = 2." Since we're told that $y = f(x)$, the derivative of y with respect to x is $\frac{dy}{dx}$, and evaluating this at x = 2 gives us $\left.\frac{dy}{dx}\right|{x=2}$, which is choice A. Choice E showing $f''(2)$ is incorrect because the double prime indicates the second derivative, not the first derivative. To master derivative notation, remember that prime notation f'(a) and Leibniz notation $\left.\frac{dy}{dx}\right|{x=a}$ both represent the same first derivative evaluated at a specific point.