Sometimes the relationship between \(x\) and \(y\) is given implicitly by an equation like \(x^2 + y^2 = 25\) rather than explicitly as \(y = f(x)\). In such cases, implicit differentiation provides a way to find \(dy/dx\). One differentiates both sides of the equation with respect to \(x\), treating \(y\) as a function of \(x\) and applying the Chain Rule whenever \(y\) appears. For example, from \(x^2 + y^2 = 25\) we get \(2x + 2y \frac{dy}{dx} = 0\), which simplifies to \(\frac{dy}{dx} = -\frac{x}{y}\). This technique is especially valuable when the equation cannot be solved explicitly for \(y\), as with curves like \(x^3 + y^3 = 6xy\), and when dealing with inverse trigonometric functions. Logarithmic differentiation is a related technique useful for complicated expressions: one takes the natural log of both sides, differentiates implicitly, and then solves for \(dy/dx\). Functions like \(y = x^x\) and products or quotients of many factors are best handled this way.
The derivative is the foundation for optimization and extremum analysis. A critical point of \(f\) occurs at \(x = c\) in the domain of \(f\) whenever \(f'(c) = 0\) or \(f'(c)\) does not exist. Every local maximum or minimum must occur at a critical point, making these the natural candidates to check. The First Derivative Test classifies a critical point: if \(f'\) changes from positive to negative at \(c\), then \(f\) has a local maximum there; if \(f'\) changes from negative to positive, then \(f\) has a local minimum; if no sign change occurs, the point is neither. The Second Derivative Test offers an alternative: if \(f'(c) = 0\) and \(f''(c) > 0\), then \(c\) is a local minimum; if \(f''(c) < 0\), then \(c\) is a local maximum. When \(f''(c) = 0\), the test is inconclusive.
The general strategy for optimization problems is to identify the quantity to optimize, express it as a function of a single variable using the given constraints, find critical points by setting the derivative to zero, and verify the result using one of the derivative tests. Related rates problems follow a similar structure: given an equation relating several time-dependent quantities, differentiate implicitly with respect to time, substitute the known rates, and solve for the unknown. For instance, if a spherical balloon's volume grows at 10 cm³/s, the formula \(V = \frac{4}{3}\pi r^3\) implies \(\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}\), so when \(r = 5\) cm, the radius increases at \(\frac{1}{10\pi} \approx 0.0318\) cm/s. In a sliding ladder problem, the Pythagorean relation \(x^2 + y^2 = 100\) yields the speed at which the top descends once the rates of one variable and the values of both are known.
Two deeper results round out the applications of derivatives. The Mean Value Theorem states that if \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there is at least one point \(c \in (a, b)\) where \(f'(c) = \frac{f(b) - f(a)}{b - a}\). Geometrically, this says that somewhere on the graph the tangent line is parallel to the secant line joining the endpoints. Concavity describes how the graph bends: \(f\) is concave up on an interval where \(f''(x) > 0\), with the graph curving upward like a cup, and concave down where \(f''(x) < 0\), with the graph curving downward like a cap. Inflection points mark where the concavity changes from one to the other, and these often coincide with points where \(f''(x) = 0\) or fails to exist.