Derivatives and Differentiation Rules

1. Introduction

The derivative is the central object of differential calculus. It answers a deceptively simple question: how fast is a quantity changing at a single instant? Velocity is the derivative of position, marginal cost is the derivative of total cost, and the slope of a tangent line is the derivative of a curve. Wherever we want to understand rates of change — in physics, economics, biology, or engineering — the derivative is the tool.

Geometrically, the derivative $f'(a)$ is the slope of the line tangent to the graph of $f$ at the point $(a, f(a))$. We obtain it by taking secant lines through nearby points and letting those points slide together, a limiting process that turns an average rate of change into an instantaneous one.

Computing derivatives from the limit definition every time would be exhausting. Fortunately, a small set of differentiation rules — the power, product, quotient, and chain rules, together with the derivatives of elementary functions — lets us differentiate almost any expression mechanically. This article develops the definition, derives the major rules with proofs, and shows how to combine them fluently.

The derivative is both a geometric object (slope of the tangent) and a physical one (instantaneous rate of change). Every differentiation rule reverses or extends the limit definition: the power rule comes from the binomial theorem, the product rule from adding a clever zero, the quotient rule from the product rule plus the chain rule on $g^{-1}$, and the chain rule from the Leibniz chain of rates. Mastering the decision tree — which rule to apply first — is as important as memorizing the formulas themselves.

2. Core Concepts

2.1 The Limit Definition of the Derivative

The derivative of $f$ at $x$ is the limit of the difference quotient: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h},$ provided this limit exists. The quotient $\frac{f(x+h)-f(x)}{h}$ is the slope of the secant line between $(x, f(x))$ and $(x+h, f(x+h))$; sending $h\to 0$ collapses the secant into the tangent. An equivalent form using a fixed point $a$ is $f'(a) = \lim_{x\to a}\frac{f(x) - f(a)}{x - a}.$

Example from definition: For $f(x) = x^2$, at any $x$: $f'(x) = \lim_{h\to 0}\frac{(x+h)^2 - x^2}{h} = \lim_{h\to 0}\frac{2xh + h^2}{h} = \lim_{h\to 0}(2x + h) = 2x.$

2.2 Differentiability and Continuity

If $f$ is differentiable at $a$, then $f$ is continuous at $a$.

Proof sketch: $\lim_{x\to a}[f(x) - f(a)] = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}\cdot(x-a) = f'(a)\cdot 0 = 0$, so $\lim_{x\to a}f(x) = f(a)$.

The converse fails: $f(x) = |x|$ is continuous at $0$ but not differentiable there (left slope $-1$, right slope $+1$). Differentiability also fails at vertical tangents (e.g. $f(x) = x^{1/3}$ at $0$) and at cusps. A differentiable function is "locally linear" — it looks like its tangent line when zoomed in.

2.3 Tangent Lines and Linear Approximation

The tangent line to $y = f(x)$ at $(a, f(a))$ has equation $y - f(a) = f'(a)(x - a), \qquad\text{or}\qquad y = L(x) = f(a) + f'(a)(x - a).$ This linearization is the best first-order approximation to $f$ near $x = a$.

2.4 Deriving the Power Rule

For a positive integer $n$, expand $(x+h)^n$ with the binomial theorem: $\frac{(x+h)^n - x^n}{h} = nx^{n-1} + \binom{n}{2}x^{n-2}h + \cdots.$ Every term after the first contains a factor of $h$, so as $h\to 0$ they vanish, leaving $f'(x) = nx^{n-1}$. The rule extends to all real exponents via implicit or logarithmic differentiation.

2.5 Deriving the Constant Multiple and Sum Rules

Constant multiple: $\frac{d}{dx}[cf(x)] = c\cdot\frac{f(x+h)-f(x)}{h} \to c f'(x)$.

Sum rule: $\frac{d}{dx}[f+g] = \lim\frac{[f(x+h)+g(x+h)]-[f(x)+g(x)]}{h} = f'(x) + g'(x)$.

2.6 Deriving the Product Rule

Let $P(x) = f(x)g(x)$. Add and subtract $f(x+h)g(x)$ in the numerator: $\frac{f(x+h)g(x+h) - f(x)g(x)}{h} = f(x+h)\frac{g(x+h)-g(x)}{h} + g(x)\frac{f(x+h)-f(x)}{h}.$ As $h\to 0$, $f(x+h)\to f(x)$ (continuity), and the difference quotients tend to $g'(x)$ and $f'(x)$, giving $(fg)' = f'g + fg'.$

2.7 Deriving the Quotient Rule

Write $\frac{f}{g} = f \cdot g^{-1}$. By the product rule and chain rule (with $(g^{-1})' = -g'/g^2$): $\left(\frac{f}{g}\right)' = f'\cdot\frac{1}{g} + f\cdot\left(-\frac{g'}{g^2}\right) = \frac{f'g - fg'}{g^2}.$

2.8 The Chain Rule

For a composite $y = f(g(x))$, write $u = g(x)$. The Leibniz form makes the idea transparent: $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}.$ Formally, $\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)$.

Proof sketch (Leibniz): If $y = f(u)$ and $u = g(x)$, then $\Delta y \approx f'(u)\,\Delta u$ and $\Delta u \approx g'(x)\,\Delta x$, so $\Delta y \approx f'(u)\,g'(x)\,\Delta x$, giving $\frac{dy}{dx} = f'(g(x))\,g'(x)$.

2.9 Derivatives of Inverse Functions

If $y = f^{-1}(x)$ and $f'(y) \neq 0$, then $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}.$ This yields $\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}$ and $\frac{d}{dx}\arctan x = \frac{1}{1+x^2}$.

2.10 Higher-Order Derivatives

The second derivative $f''(x) = (f')'(x)$ measures the rate of change of the slope (concavity). The $n$th derivative $f^{(n)}(x)$ is defined inductively. Notation: $f''$, $f'''$, $f^{(n)}$, or $\frac{d^n y}{dx^n}$.

2.11 Implicit and Logarithmic Differentiation

When $y$ is defined implicitly by $F(x,y) = 0$, differentiate both sides with respect to $x$, treating $y$ as a function of $x$ (chain rule on every $y$-term), then solve for $y'$.

For $y = f(x)^{g(x)}$ or products of many factors, take $\ln$ of both sides: $\ln y = g(x)\ln f(x)$, differentiate implicitly, then solve for $y'$.

2.12 Mean Value Theorem Preview

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then some $c \in (a,b)$ satisfies $f'(c) = \frac{f(b)-f(a)}{b-a}$. Differentiability is what connects local slopes to global average change — a theme developed fully in applications of derivatives.

2.13 Derivatives of Trigonometric Functions from First Principles

Using $\lim_{h\to 0}\frac{\sin h}{h} = 1$ and the angle-addition formula: $\frac{d}{dx}\sin x = \lim_{h\to 0}\frac{\sin(x+h)-\sin x}{h} = \lim_{h\to 0}\cos x\cdot\frac{\sin h}{h} + \sin x\cdot\frac{\cos h - 1}{h} = \cos x.$ Similarly, $\frac{d}{dx}\cos x = -\sin x$ follows from the cosine addition formula.

2.14 The Derivative as a Linear Operator

Differentiation is linear: $(cf + dg)' = cf' + dg'$ for constants $c, d$. This means we can differentiate term-by-term in any finite sum. It does not mean $(fg)' = f'g'$ — products require the product rule.

2.15 Parametric and Related Rates Preview

If $x = x(t)$ and $y = y(t)$, the chain rule gives $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ when $\frac{dx}{dt} \neq 0$. This is the foundation of related rates and parametric tangent slopes.