Differential Calculus of One Variable

This chapter systematically studies the derivatives and differentials of single-variable functions, including their definitions, geometric and physical meanings, basic differentiation rules, derivatives of common functions, mean value theorems, L’Hospital’s rule, and the application of derivatives in studying function properties (such as monotonicity, extrema, concavity, inflection points, and maximum/minimum values).

Basic Concepts of Derivatives and Differentials

Definition of Derivative

Let $y = f(x)$ be defined in a neighborhood of $x_0$. If the limit

exists, it is called the derivative of $f(x)$ at $x_0$, denoted as $f'(x_0)$ or $\frac{dy}{dx}\bigg|_{x_0}$.

Geometric Meaning of Derivative

The derivative represents the slope of the tangent to the curve $y = f(x)$ at $x_0$.

Physical Meaning of Derivative

In physics, the derivative often represents the instantaneous rate of change, such as velocity being the derivative of displacement with respect to time.

Differentiability and Continuity

If a function is differentiable at a point, it must be continuous there; but continuity does not necessarily imply differentiability (e.g., $|x|$ at $x=0$).

Definition of Differential

If $f(x)$ is differentiable at $x$, then $df = f'(x)dx$, where $df$ is called the differential of $f(x)$ at $x$.

Basic Differentiation Rules and Common Formulas

• Sum and difference rule: $(u \pm v)' = u' \pm v'$
• Product rule: $(uv)' = u'v + uv'$
• Quotient rule: $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$
• Chain rule (composite function): $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Derivatives of common functions:

• Power function: $(x^n)' = n x^{n-1}$
• Exponential function: $(a^x)' = a^x \ln a$
• Logarithmic function: $(\ln x)' = \frac{1}{x}$
• Trigonometric functions: $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$, $(\tan x)' = \sec^2 x$

Differentiation of Composite, Inverse, Implicit, and Higher-Order Derivatives

• Composite function: chain rule
• Inverse function: the derivative of $f^{-1}(x)$ is $= \frac{1}{f'(f^{-1}(x))}$
• Implicit function: differentiate both sides of the equation with respect to $x$
• Parametric equations: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$
• Higher-order derivatives: $f''(x)$, $f'''(x)$, etc.

Mean Value Theorems

• Rolle’s Theorem: If $f(a) = f(b)$, $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $\xi \in (a, b)$ such that $f'(\xi) = 0$
• Lagrange Mean Value Theorem: There exists $\xi \in (a, b)$ such that $f'(\xi) = \frac{f(b) - f(a)}{b - a}$
• Cauchy’s Mean Value Theorem: If $g'(x) \neq 0$, there exists $\xi \in (a, b)$ such that $\frac{f'(\xi)}{g'(\xi)} = \frac{f(b) - f(a)}{g(b) - g(a)}$

L’Hospital’s Rule

If $\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0$ or $\pm\infty$, and $g'(x) \neq 0$, then

(if the limit on the right exists)

Applications of Derivatives

Monotonicity and Extrema

• $f'(x) > 0$: $f(x)$ is increasing; $f'(x) < 0$: decreasing
• $f'(x_0) = 0$ and $f''(x_0) > 0$: $x_0$ is a minimum point; $f''(x_0) < 0$: maximum point

Concavity and Inflection Points

• $f''(x) > 0$: graph is concave up; $f''(x) < 0$: concave down
• $f''(x_0) = 0$ and the second derivative changes sign: $x_0$ is an inflection point

Maximum and Minimum Values

• On a closed interval $[a, b]$, the maximum/minimum may occur at endpoints or stationary points

Tangents and Normals to Curves

• Slope of tangent $k = f'(x_0)$, equation: $y - f(x_0) = f'(x_0)(x - x_0)$
• Slope of normal $-1/f'(x_0)$

Arc Differential and Curvature

• Arc differential $ds = \sqrt{1 + [f'(x)]^2} dx$
• Curvature $\kappa = \frac{|f''(x)|}{[1 + (f'(x))^2]^{3/2}}$

Exercises

1. Find the extrema and extreme values of $f(x) = x^3 - 3x^2 + 2$.
2. Let $y = x^x$, find $y'$.
3. Find the equation of the tangent to the curve $y = \ln x$ at the point $(1, 0)$.
4. Use L’Hospital’s rule to compute $\lim\limits_{x \to 0} \frac{\sin x}{x}$.
5. Determine the differentiability and continuity of $f(x) = |x|$ at $x = 0$.
6. Let $f(x)$ be continuous on $[a, b]$ and differentiable on $(a, b)$, with $f(a) = f(b)$. Prove that there exists $\xi \in (a, b)$ such that $f'(\xi) = 0$.