Single-Variable

The foundation of calculus begins with a single variable—exploring how one quantity changes with respect to another. Single-variable calculus builds intuition for limits, derivatives, and integrals, laying the groundwork for deeper mathematical modeling and real-world applications. 📈

Learning Objectives

Learning objectives of the Single-Variable section.

• Derivatives
• Integrals
• Fundamental Theorem of Calculus
• Optimization

Summary Table

Summary of the Single-Variable section.

Concept	Description	Notation
Derivative	Instantaneous rate of change; slope of the tangent line	f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
Constant Rule	Derivative of a constant function	\frac{d}{dx}[c] = 0
Power Rule	Derivative of a power function	\frac{d}{dx}[x^n] = nx^{n-1}
Sum/Difference Rule	Derivative of a sum or difference	\frac{d}{dx}[g(x) \pm h(x)] = g'(x) \pm h'(x)
Product Rule	Derivative of a product	\frac{d}{dx}[g(x)h(x)] = g'(x)h(x) + g(x)h'(x)
Quotient Rule	Derivative of a quotient	\frac{d}{dx}\left[\frac{g(x)}{h(x)}\right] = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}
Chain Rule	Derivative of a composite function	\frac{d}{dx}[g(h(x))] = g'(h(x)) \cdot h'(x)
Definite Integral	Net area under a curve from a to b	\int_a^b f(x)\, dx
Indefinite Integral	General antiderivative	\int f(x)\, dx = F(x) + C
Constant Rule (Int.)	Integral of a constant	\int c\,dx = cx + C
Power Rule (Int.)	Integral of a power function (when n \neq -1)	\int x^n\,dx = \frac{x^{n+1}}{n+1} + C
Sum/Difference Rule (Int.)	Integral of a sum or difference	\int [g(x) \pm h(x)]\,dx = \int g(x)\,dx \pm \int h(x)\,dx
Exponential Rule (base e)	Integral of e^x	\int e^x\,dx = e^x + C
Exponential Rule (base a)	Integral of a^x	\int a^x\,dx = \frac{a^x}{\ln a} + C
Logarithmic Rule	Integral of reciprocal	\int \frac{1}{x}\,dx = \ln|x| + C
Chain Rule (u-sub)	Substitution technique	\int f(g(x))g'(x)\,dx = \int f(u)\,du,\ u = g(x)
Integration by Parts	Product of functions technique	\int u(x)v'(x)\,dx = u(x)v(x) - \int u'(x)v(x)\,dx
Fundamental Theorem	Derivative of accumulation function	F(x) = \int_a^x f(t)\,dt \Rightarrow F'(x) = f(x)
FTC Evaluation	Total accumulation from a to b	\int_a^b f(x)\,dx = F(b) - F(a)
Optimization	Find extrema using derivatives	f'(x) = 0, f''(x) \gtrless 0
First Derivative Test	Sign change test for local extrema	f'(x): + \to - (max), - \to + (min), no change = none
Second Derivative Test	Concavity test for local extrema	f''(x) > 0: min, f''(x) < 0: max, f''(x) = 0: inconclusive