Applications of Derivatives

1. Introduction

Once you can compute derivatives, the real payoff begins: the derivative becomes a lens for analyzing and optimizing the behavior of functions and the real-world systems they model. Because $f'(x)$ measures the instantaneous rate of change, its sign tells us when a quantity is rising or falling, its zeros locate peaks and valleys, and the second derivative reveals the curvature that distinguishes a maximum from a minimum.

This article covers the major applications of the derivative in a first calculus course: classifying increasing/decreasing behavior and concavity, finding local and absolute extrema, solving optimization problems, modeling related rates, using linear approximation, and applying the Mean Value Theorem (MVT) and Rolle's Theorem to connect average and instantaneous rates.

The common thread is translation: a verbal or geometric situation becomes an equation, the derivative extracts the information we need, and we interpret the result back in context. Mastering this translation is the most valuable skill in applied calculus.

The theoretical backbone of this chapter is the Mean Value Theorem, which follows from Rolle's Theorem, which in turn rests on the Extreme Value Theorem and Fermat's observation that interior extrema have zero derivative. This chain of results justifies the First Derivative Test (via monotonicity from $f' > 0$), supports error bounds for linear approximation, and connects to L'Hôpital's rule for indeterminate limits. Applied problems — optimization and related rates — require careful setup before any differentiation occurs.

2. Core Concepts

2.1 Increasing/Decreasing and the First Derivative Test

On an interval, $f$ is increasing where $f'(x) > 0$ and decreasing where $f'(x) < 0$. Points where $f'(x) = 0$ or $f'$ is undefined are critical points — the only candidates for local extrema.

First Derivative Test: At a critical point $c$, if $f'$ changes from $+$ to $-$, then $c$ is a local maximum; from $-$ to $+$, a local minimum; if $f'$ does not change sign, $c$ is neither.

Proof sketch (MVT connection): If $f'(x) > 0$ on $(a,b)$, then for any $x_1 < x_2$ in the interval, the MVT gives $f(x_2) - f(x_1) = f'(c)(x_2 - x_1) > 0$, so $f$ is increasing.

2.2 Concavity and the Second Derivative Test

Where $f''(x) > 0$ the graph is concave up (slopes increasing); where $f''(x) < 0$ it is concave down. An inflection point is where concavity changes (and $f$ is continuous).

Second Derivative Test: If $f'(c) = 0$ and $f''(c) > 0$, then $c$ is a local minimum; if $f''(c) < 0$, a local maximum; if $f''(c) = 0$, inconclusive — use the First Derivative Test.

Proof sketch (local min): If $f''(c) > 0$, then $f'$ is increasing near $c$, so $f'$ is negative just left of $c$ and positive just right — a sign change from $-$ to $+$, confirming a local minimum.

2.3 Absolute Extrema and the Extreme Value Theorem

A function continuous on a closed interval $[a,b]$ attains both an absolute maximum and minimum (EVT). These occur at critical points inside $(a,b)$ or at endpoints. The Closed Interval Method evaluates $f$ at every critical point and both endpoints, then selects the largest and smallest values.

2.4 Rolle's Theorem

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists $c \in (a,b)$ with $f'(c) = 0$.

Proof sketch: By the EVT, $f$ attains a max or min at some $c \in (a,b)$ (since $f(a) = f(b)$, an interior extremum exists unless $f$ is constant). At that interior extremum, $f'(c) = 0$ by Fermat's theorem on interior extrema.

2.5 The Mean Value Theorem (MVT)

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ with $f'(c) = \frac{f(b) - f(a)}{b - a}.$

Proof sketch: Define $g(x) = f(x) - \frac{f(b)-f(a)}{b-a}(x - a)$. Then $g(a) = g(b)$, so Rolle's theorem gives $g'(c) = 0$, i.e. $f'(c) = \frac{f(b)-f(a)}{b-a}$.

Geometrically, some tangent is parallel to the secant through the endpoints. Physically, instantaneous speed equals average speed at some moment.

2.6 Linearization and Differentials

The linearization near $x = a$ is $L(x) = f(a) + f'(a)(x - a)$. The differential $dy = f'(x)\,dx$ estimates the change in $y$ for a small change $dx$. The error in linear approximation is $O((x-a)^2)$ when $f''$ is bounded.

2.7 Related Rates

When quantities vary in time and are linked by an equation, differentiating with respect to $t$ (chain rule) relates their rates. Structure: constraint $\Rightarrow$ differentiate in $t$ $\Rightarrow$ substitute instantaneous values $\Rightarrow$ solve for the unknown rate.

2.8 Newton's Method (Optional)

To approximate a root of $f(x) = 0$, iterate $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}.$ This follows from linearizing $f$ at $x_n$ and solving for the $x$-intercept of the tangent line. Convergence is fast near simple roots but not guaranteed globally.

2.9 L'Hôpital's Rule Connection

Indeterminate limits $\tfrac{0}{0}$ and $\tfrac{\infty}{\infty}$ can be resolved by $\lim \frac{f}{g} = \lim \frac{f'}{g'}$, which is essentially an MVT/Cauchy MVT consequence — the ratio of function values equals the ratio of derivatives at some intermediate point.

2.10 Curve Sketching: Asymptotes from Derivatives

Vertical asymptotes occur where the function blows up (often zeros of denominators). Horizontal/oblique asymptotes come from limits at $\pm\infty$. Combining asymptote analysis with $f'$ and $f''$ sign charts produces complete graphs without a calculator.

2.11 Marginal Analysis in Economics

If $C(x)$ is total cost and $R(x)$ is revenue, then $C'(x)$ is marginal cost and $R'(x)$ is marginal revenue. Profit is maximized when marginal revenue equals marginal cost: $R'(x) = C'(x)$, or equivalently when the marginal profit $P'(x) = R'(x) - C'(x) = 0$.

2.12 Error Bounds for Linear Approximation

If $|f''(x)| \le M$ near $a$, the error in $L(x) = f(a) + f'(a)(x-a)$ satisfies $|f(x) - L(x)| \le \frac{M}{2}(x-a)^2.$ This is a direct consequence of Taylor's theorem with $N = 1$.