Essential Calculus Theorems and Formula Reference

Posted on Nov 25, 2025 in Mathematics

Fundamental Calculus Definitions and Theorems

The Derivative and Integral Definitions

The Derivative Definition

The derivative of a function $f(x)$, denoted $f'(x)$, is defined using the limit of the difference quotient:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) – f(x)}{h}$$

The Definite Integral (Riemann Sum)

The definite integral of $f(x)$ from $a$ to $b$ is defined as the limit of the Riemann sum:

$$\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$$

Key Calculus Theorems

Mean Value Theorem (MVT)

  • Conditions:
    • The function $f(x)$ must be continuous on the closed interval $[a,b]$.
    • The function $f(x)$ must be differentiable on the open interval $(a,b)$.
  • Conclusion: There exists a number $c$ in $(a,b)$ such that the instantaneous rate of change equals the average rate of change: $$f'(c) = \frac{f(b) – f(a)}{b – a}$$

Intermediate Value Theorem (IVT)

  • Condition: The function $f(x)$ must be continuous on the closed interval $[a,b]$.
  • Conclusion: The function takes on every value between $f(a)$ and $f(b)$ at some point within the interval $[a,b]$.

L’Hôpital’s Rule

L’Hôpital’s Rule is used to evaluate indeterminate forms ($\frac{0}{0}$ or $\frac{\pm\infty}{\pm\infty}$).

  • Conditions: Functions $f(x)$ and $g(x)$ must be differentiable on an open interval containing $a$ (except possibly at $a$).
  • Conclusion: The limit of the quotient of the derivatives can be computed by taking the derivative of the numerator and denominator separately: $$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$

Essential Differentiation Formulas

Below are the fundamental rules for finding derivatives:

  • Constant Rule: $\frac{d}{dx}(c) = 0$, where $c$ is a constant.
  • Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n$ is a real number.
  • Sum Rule: $\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$.
  • Difference Rule: $\frac{d}{dx}(f(x) – g(x)) = f'(x) – g'(x)$.
  • Product Rule: $\frac{d}{dx}(f(x) \cdot g(x)) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$.
  • Quotient Rule: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x) \cdot g(x) – f(x) \cdot g'(x)}{(g(x))^2}$.
  • Chain Rule: $\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$.

Derivatives of Specific Functions

  • Exponential Function: $\frac{d}{dx}(e^x) = e^x$.
  • Natural Logarithm: $\frac{d}{dx}(\ln(x)) = \frac{1}{x}$.
  • Logarithmic Functions (Base $a$): $\frac{d}{dx}(\log_a(x)) = \frac{1}{x \cdot \ln(a)}$.
  • General Logarithm: $\frac{d}{dx}(\log(x)) = \frac{1}{x}$. (This formula assumes $\log(x)$ is the natural logarithm $\ln(x)$.)

Trigonometric Derivatives

  • $\frac{d}{dx}(\sin(x)) = \cos(x)$
  • $\frac{d}{dx}(\cos(x)) = -\sin(x)$
  • $\frac{d}{dx}(\tan(x)) = \sec^2(x)$

Inverse Trigonometric Derivatives

  • $\frac{d}{dx}(\arcsin(x)) = \frac{1}{\sqrt{1 – x^2}}$
  • $\frac{d}{dx}(\arccos(x)) = -\frac{1}{\sqrt{1 – x^2}}$
  • $\frac{d}{dx}(\arctan(x)) = \frac{1}{1 + x^2}$

Hyperbolic Derivatives

  • $\frac{d}{dx}(\sinh(x)) = \cosh(x)$
  • $\frac{d}{dx}(\cosh(x)) = \sinh(x)$
  • $\frac{d}{dx}(\tanh(x)) = \text{sech}^2(x)$

Inverse Hyperbolic Derivatives

  • $\frac{d}{dx}(\text{arcsinh}(x)) = \frac{1}{\sqrt{x^2 + 1}}$
  • $\frac{d}{dx}(\text{arccosh}(x)) = \frac{1}{\sqrt{x^2 – 1}}$
  • $\frac{d}{dx}(\text{arctanh}(x)) = \frac{1}{1 – x^2}$

Chain Rule Applications

  • Exponential Function: $\frac{d}{dx}(a^{f(x)}) = \ln(a) \cdot a^{f(x)} \cdot f'(x)$, where $a$ is a constant.
  • Natural Logarithm: $\frac{d}{dx}(\ln(f(x))) = \frac{f'(x)}{f(x)}$.
  • Trigonometric (Sine): $\frac{d}{dx}(\sin(f(x))) = \cos(f(x)) \cdot f'(x)$.
  • Trigonometric (Cosine): $\frac{d}{dx}(\cos(f(x))) = -\sin(f(x)) \cdot f'(x)$.
  • Trigonometric (Tangent): $\frac{d}{dx}(\tan(f(x))) = \sec^2(f(x)) \cdot f'(x)$.
  • Inverse Trigonometric (Arcsine): $\frac{d}{dx}(\arcsin(f(x))) = \frac{f'(x)}{\sqrt{1 – (f(x))^2}}$.
  • Inverse Trigonometric (Arccosine): $\frac{d}{dx}(\arccos(f(x))) = -\frac{f'(x)}{\sqrt{1 – (f(x))^2}}$.
  • Inverse Trigonometric (Arctangent): $\frac{d}{dx}(\arctan(f(x))) = \frac{f'(x)}{1 + (f(x))^2}$.

Essential Integration Formulas

Below are the fundamental rules for finding antiderivatives (Indefinite Integrals), where $C$ is the constant of integration:

  • Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$.
  • Constant Rule: $\int c dx = cx + C$, where $c$ is a constant.
  • Sum Rule: $\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$.
  • Difference Rule: $\int (f(x) – g(x)) dx = \int f(x) dx – \int g(x) dx$.
  • Exponential Function: $\int e^x dx = e^x + C$.
  • Natural Logarithm: $\int \frac{1}{x} dx = \ln|x| + C$.

Trigonometric Integrals

  • $\int \sin(x) dx = -\cos(x) + C$
  • $\int \cos(x) dx = \sin(x) + C$
  • $\int \sec^2(x) dx = \tan(x) + C$

Inverse Trigonometric Integrals

  • $\int \frac{1}{\sqrt{1-x^2}} dx = \arcsin(x) + C$
  • $\int \frac{-1}{\sqrt{1-x^2}} dx = \arccos(x) + C$
  • $\int \frac{1}{1+x^2} dx = \arctan(x) + C$

Logarithmic and Hyperbolic Integrals

  • Logarithmic (Base $a$): $\int \frac{1}{x \ln(a)} dx = \frac{\ln|x|}{\ln(a)} + C$.
  • Hyperbolic Sine: $\int \sinh(x) dx = \cosh(x) + C$
  • Hyperbolic Cosine: $\int \cosh(x) dx = \sinh(x) + C$
  • Hyperbolic Secant Squared: $\int \text{sech}^2(x) dx = \tanh(x) + C$

Inverse Hyperbolic Integrals

  • $\int \frac{1}{\sqrt{x^2 + 1}} dx = \text{arcsinh}(x) + C$
  • $\int \frac{1}{\sqrt{x^2 – 1}} dx = \text{arccosh}(x) + C$
  • $\int \frac{1}{1 – x^2} dx = \text{arctanh}(x) + C$

The Fundamental Theorem of Calculus (FTC)

The FTC establishes the crucial connection between differentiation and integration.

FTC Part 1

It states that if $f$ is continuous on $[a,b]$, then the function $g(x)$ defined by $g(x) = \int_a^x f(t) dt$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and $g'(x) = f(x)$.

FTC Part 2

It states that if $f$ is continuous on $[a,b]$ and $F$ is any antiderivative of $f$ (i.e., $F'(x) = f(x)$), then the definite integral is calculated by evaluating the antiderivative at the endpoints:

$$\int_a^b f(x) dx = F(b) – F(a)$$

In summary, the FTC allows us to evaluate definite integrals and find derivatives of functions defined as integrals.

Average and Instantaneous Rates

Average Rate of Change (ARC)

The average rate of change of $f(x)$ over the interval $[a,b]$ is the slope of the secant line:

$$\text{ARC} = \frac{f(b) – f(a)}{b – a}$$

Average Value of a Function ($f_{\text{avg}}$)

The average value of a continuous function $f(x)$ on the interval $[a,b]$ is given by:

$$f_{\text{avg}} = \frac{1}{b – a} \int_a^b f(x) dx$$