Comprehensive Reference Guide to Differential Calculus: Definitions, Rules, and Elementary Derivatives

Geometric Interpretation and Limit Definition of the Derivative

• Analytical Limit Definition:

  • The derivative of a function $f(x)$ is formally defined as the limit of the difference quotient as the increment $h$ approaches zero:
  • $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

• Geometric Meaning:

  • The derivative $f'(x)$ evaluated at a given point represents the exact slope of the tangent line to the curve of the function at that specific point.

Fundamental Operational Rules of Differentiation

• Sum Rule (Suma):

  • The derivative of the sum of two differentiable functions is the sum of their individual derivatives.
  • Given $F(x) = f(x) + g(x)$, the derivative is:
  • $F'(x) = f'(x) + g'(x)$

• Constant Multiple Rule (Producte per un nombre):

  • The derivative of a constant multiplied by a differentiable function is equal to the constant multiplied by the derivative of the function.
  • Given $F(x) = k \cdot f(x)$, where $k$ is a constant scalar, the derivative is:
  • $F'(x) = k \cdot f'(x)$

• Product Rule (Multiplicació):

  • The derivative of the product of two functions equals the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.
  • Given $F(x) = f(x) \cdot g(x)$, the derivative is:
  • $F'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$

• Quotient Rule (Divisió):

  • The derivative of the ratio of two differentiable functions is equal to the derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator, all divided by the denominator squared.
  • Given $F(x) = \frac{f(x)}{g(x)}$, where $g(x) \neq 0$, the derivative is:
  • $F'(x) = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2}$

• Chain Rule for Composite Functions (Regla de la cadena):

  • Composition of Two Functions:
  • For a composite function where an outer function encompasses an inner function, the derivative is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
  • Given $F(x) = f(g(x))$, the derivative is:
  • $F'(x) = f'(g(x)) \cdot g'(x)$
  • Composition of Three Nested Functions:
  • When three functions are nested inside one another, the chain rule extends sequentially from the outermost function to the innermost function.
  • Given $F(x) = h(g(f(x)))$, the derivative is:
  • $F'(x) = h'(g(f(x))) \cdot g'(f(x)) \cdot f'(x)$

Differentiation of Power and Algebraic Functions

• General Power Rule (Potència):

  • For any variable raised to a real exponent $n$, the derivative is found by multiplying by the exponent and reducing the power by one.
  • Given $f(x) = x^n$, the derivative is:
  • $f'(x) = n \cdot x^{n - 1}$

• Square Root Function:

  • The square root is expressed algebraically as a fractional power $f(x) = \sqrt{x} = x^{\frac{1}{2}}$.
  • Differentiating using the power rule yields:
  • $f'(x) = \frac{1}{2\sqrt{x}}$

• Reciprocal Function:

  • The reciprocal of $x$ is expressed as a negative power $f(x) = \frac{1}{x} = x^{-1}$.
  • Differentiating using the power rule yields:
  • $f'(x) = -\frac{1}{x^2}$

Differentiation of Exponential and Logarithmic Functions

• Exponential Functions (Exponencials):

  • Natural Exponential Function:
  • The exponential function with the natural base $e$ reproduces itself under differentiation.
  • Given $f(x) = e^x$, the derivative is:
  • $f'(x) = e^x$
  • General Base Exponential Function:
  • For an exponential function with an arbitrary positive base $a$ (where $a > 0$ and $a \neq 1$), the derivative is the original function scaled by the natural logarithm of the base.
  • Given $f(x) = a^x$, the derivative is:
  • $f'(x) = a^x \cdot \ln(a)$

• Logarithmic Functions (Logarítmiques):

  • Natural Logarithm with Positive Argument:
  • For $f(x) = \ln(x)$, restricted to the domain where $x > 0$, the derivative is:
  • $f'(x) = \frac{1}{x}$
  • Natural Logarithm with Absolute Value Argument:
  • For $f(x) = \ln(|x|)$, defined for all non-zero real values where $x \neq 0$, the derivative is:
  • $f'(x) = \frac{1}{x}$
  • General Base Logarithm with Positive Argument:
  • For a logarithm with an arbitrary base $a$ (where $a > 0$ and $a \neq 1$) defined for $x > 0$:
  • Given $f(x) = \log_a(x)$, the derivative is:
  • $f'(x) = \frac{1}{x \cdot \ln(a)}$
  • General Base Logarithm with Absolute Value Argument:
  • For a logarithm with base $a$ defined for all non-zero values where $x \neq 0$:
  • Given $f(x) = \log_a(|x|)$, the derivative is:
  • $f'(x) = \frac{1}{x \cdot \ln(a)}$

Differentiation of Trigonometric and Inverse Trigonometric Functions

• Trigonometric Functions (Trigonomètriques):

  • Sine Function:
  • Given $f(x) = \sin(x)$, the derivative is:
  • $f'(x) = \cos(x)$
  • Cosine Function:
  • Given $f(x) = \cos(x)$, the derivative is:
  • $f'(x) = -\sin(x)$
  • Tangent Function:
  • Given $f(x) = \tan(x)$, the derivative can be written in two equivalent standard mathematical forms:
  • $f'(x) = 1 + \tan^2(x)$
  • $f'(x) = \frac{1}{\cos^2(x)}$

• Inverse Trigonometric Functions (Funcions arc):

  • Arcsine Function:
  • Given $f(x) = \arcsin(x)$, the derivative is:
  • $f'(x) = \frac{1}{\sqrt{1 - x^2}}$
  • Arccosine Function:
  • Given $f(x) = \arccos(x)$, the derivative is:
  • $f'(x) = -\frac{1}{\sqrt{1 - x^2}}$
  • Arctangent Function:
  • Given $f(x) = \arctan(x)$, the derivative is:
  • $f'(x) = \frac{1}{1 + x^2}$