Derivative: Definition and Geometric Meaning

History: Newton and Leibniz's Disputes

At the end of the 17th century, Newton and Leibniz independently developed differential calculus. Newton called the derivative "fluxion" and thought of it physically—as instantaneous velocity. Leibniz devised the convenient notation dy/dx, which we still use today. Their approaches were equivalent, but the years-long feud over priority poisoned the mathematical community for decades.

A rigorous definition of the derivative was given by Cauchy in the 19th century.

Definition of the Derivative

The derivative of a function f at the point $x_0$ is the limit of the ratio of the increment of the function to the increment of the argument:

$ f'(x_0) = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} $

If this limit exists, the function is called differentiable at the point $x_0$.

Physical meaning: $f'(x_0)$ is the instantaneous rate of change of $f$ at $x_0$. If $f(t)$ is the position of a particle, then $f'(t)$ is its velocity.

Geometric meaning: $f'(x_0)$ is the slope of the tangent to the graph of $f$ at the point $(x_0, f(x_0))$.

Connection Between Differentiability and Continuity

Differentiability $\rightarrow$ continuity. The converse is not true.

The function $|x|$ is continuous at zero, but not differentiable: the left derivative is $-1$, the right is $1$. A "corner" on the graph is a point of non-differentiability.

The Weierstrass function—a function that is continuous but nowhere differentiable—showed that "infinitely angular" functions exist. This was a sensation of the 19th century.

Differentiation Rules

$(cf)' = cf'$, $(f+g)' = f'+g'$, $(fg)' = f'g + fg'$, $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$.

Derivative of a composite function: $(f(g(x)))' = f'(g(x)) \cdot g'(x)$.

Table of basic derivatives:

• $(x^n)' = nx^{n-1}$
• $(e^x)' = e^x$
• $(\ln x)' = 1/x$
• $(\sin x)' = \cos x$
• $(\cos x)' = -\sin x$
• $(\arctan x)' = 1/(1+x^2)$

Higher Order Derivatives

$f''(x)$ is the derivative of the derivative. Physically: acceleration. Geometrically: related to convexity.

$f''(x_0) > 0$—the function is convex downward at the point $x_0$. $f''(x_0) < 0$—convex upward.

Differential

The differential $df = f'(x_0) \cdot dx$ is the linear part of the increment of the function. Increment: $\Delta f = f'(x_0)\Delta x + o(\Delta x)$. The differential is the main linear part of the increment.

The differential allows approximate calculations: $\sqrt{4.01} \approx \sqrt{4} + \frac{1}{2\sqrt{4}} \cdot 0.01 = 2 + 0.0025 = 2.0025$.

The Inverse Function Theorem

If $f$ is differentiable at the point $x_0$ and $f'(x_0) \neq 0$, then the inverse function $f^{-1}$ exists in a neighborhood of $y_0 = f(x_0)$ and is differentiable:

$ (f^{-1})'(y_0) = \frac{1}{f'(x_0)} $

Geometrically: the slope of the tangent to the inverse function is the reciprocal of the slope of the original. If the tangent to $f$ is not horizontal ($f'(x_0) \neq 0$), the inverse function is also differentiable.

Example: $(\arcsin x)'$ at $x = \sin t$: $d(\arcsin x)/dx = 1/(\sin'(t)) = 1/\cos(t) = 1/\sqrt{1 − x^2}$. This is the standard formula for the derivative of arcsine.

Logarithmic Differentiation

If a function is given as the product of many factors or contains a variable in the exponent, it is convenient to logarithmize: instead of $f(x)$, differentiate $\ln f(x)$.

Example: $f(x) = x^x$. $\ln f = x \ln x$. Differentiating: $f'/f = \ln x + 1$, hence $f' = x^x(\ln x + 1)$.

Applications in Physics and Economics

Velocity and acceleration are the first and second derivatives of position with respect to time. Marginal revenue $MR = dTR/dQ$ is the derivative of total revenue with respect to quantity. The condition for maximized profit $MR = MC$ is derived by equating derivatives. Elasticity of demand $E = (dQ/dp) \cdot (p/Q)$ is the derivative, normalized to the level: shows by what percent demand will change when price is changed by 1%.

Numerical Differentiation

When a function is given in tabular form, the derivative is approximated by finite differences. First-order formula: $f'(x) \approx (f(x+h) − f(x))/h$. Second-order formula: $f'(x) \approx (f(x+h) − f(x−h))/(2h)$—"central difference," more accurate for the same step $h$. The error is estimated using Taylor's formula.

Derivative as a Linear Mapping

In the multivariate generalization, the derivative of a function $f: \mathbb{R}^n \to \mathbb{R}^m$ is not a number, but a linear mapping (Jacobian matrix) that best approximates $f$ near a given point. For a function $f: \mathbb{R} \to \mathbb{R}$ the Jacobian matrix is simply a $1 \times 1$ matrix, that is, the number $f'(x)$. This viewpoint unites differentiation of functions of one and many variables in a single theory.

Question for reflection: The Weierstrass function $f(x) = \Sigma a^n\cos(b^n\pi x)$ (where $0 < a < 1$, $b$ is an odd integer, $ab > 1 + 3\pi/2$) is everywhere continuous but nowhere differentiable. How is this possible intuitively?

Derivative of the Inverse Function and Logarithmic Differentiation

If $f$ is monotonic and differentiable at the point $x$, then $(f^{-1})'(y) = 1/f'(x)$ with $y = f(x)$. Example: $(\arcsin)'(y) = 1/\cos(\arcsin y) = 1/\sqrt{1 − y^2}$. Logarithmic differentiation: for $y = u(x)^{v(x)}$ take $\ln y = v \ln u$, differentiate: $y'/y = v' \ln u + v u'/u$. Corollary: $(x^x)' = x^x(1 + \ln x)$. This technique allows one to differentiate products of many functions ($\ln |\prod f_i| = \Sigma \ln|f_i|$) and functions with variable base and exponent—a standard in theoretical physics and in proofs of inequalities.