Key Points

The derivative $\frac{dy}{dx}$ represents the instantaneous rate of change of a quantity $y$ with respect to another quantity $x$. For a function $y = f(x)$, the rate of change at a specific point $x=x_0$ is given by the value of the derivative at that point, $f'(x_0)$.

If two variables $x$ and $y$ both vary with respect to a third variable, typically time $t$, their rates of change are related by the Chain Rule. The formula is $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}$, which allows finding one rate when others are known.

A function $f(x)$ is increasing on an interval $(a, b)$ if its derivative $f'(x) \geq 0$ for all $x$ in that interval. The function is strictly increasing if $f'(x) > 0$ for all $x$ in $(a, b)$.

A function $f(x)$ is decreasing on an interval $(a, b)$ if its derivative $f'(x) \leq 0$ for all $x$ in that interval. The function is strictly decreasing if $f'(x) < 0$ for all $x$ in $(a, b)$.

To find the intervals where a function is increasing or decreasing, first find the critical points by solving $f'(x) = 0$. These points divide the number line into intervals. Test the sign of $f'(x)$ in each interval to determine the function's behavior.

The slope of the tangent to a curve $y = f(x)$ at a point $(x_1, y_1)$ is given by the derivative $m = f'(x_1)$. The equation of the tangent line is then found using the point-slope form: $y - y_1 = m(x - x_1)$.

The normal line is perpendicular to the tangent at the point of contact. Its slope is the negative reciprocal of the tangent's slope, $m_{\text{normal}} = -\frac{1}{f'(x_1)}$, provided $f'(x_1) \neq 0$. The equation is $y - y_1 = -\frac{1}{f'(x_1)}(x - x_1)$.

A critical point of a function $f(x)$ is a point $c$ in its domain where either the derivative is zero, $f'(c) = 0$, or the derivative is not defined. Local maxima or minima can only occur at these critical points.

At a critical point $c$: if $f'(x)$ changes sign from positive to negative, $c$ is a point of local maximum. If $f'(x)$ changes sign from negative to positive, $c$ is a point of local minimum. If there is no sign change, it is a point of inflection.

Let $c$ be a critical point where $f'(c) = 0$. If the second derivative $f''(c) < 0$, then $c$ is a point of local maximum. If $f''(c) > 0$, then $c$ is a point of local minimum. If $f''(c) = 0$, the test is inconclusive.

To find the absolute maximum and minimum of a function $f(x)$ on a closed interval $[a, b]$, find all critical points in $(a, b)$. Evaluate $f(x)$ at these critical points and also at the endpoints $a$ and $b$. The largest of these values is the absolute maximum, and the smallest is the absolute minimum.

In economics, if $C(x)$ is the total cost of producing $x$ units, the marginal cost is $MC = \frac{dC}{dx}$. If $R(x)$ is the total revenue from selling $x$ units, the marginal revenue is $MR = \frac{dR}{dx}$.