Differentiation in Calculus

Differentiation

• Differentiation is the process of finding the derivative of a function.
• The derivative of a function represents the instantaneous rate of change of the function at a given point.
• It is often visualized as the slope of the tangent line to the function at that point.
• Common rules for differentiation include:
  • Power rule: The derivative of xⁿ is nx^n-1
  • Constant multiple rule: The derivative of cf(x) is cf'(x)
  • Sum/difference rule: The derivative of (f(x) ± g(x)) is f'(x) ± g'(x)
  • Product rule: The derivative of (f(x)*g(x)) is f'(x)g(x) + f(x)g'(x)
  • Quotient rule: The derivative of (f(x)/g(x)) is (f'(x)g(x) - f(x)g'(x)) / (g(x))²
  • Chain rule: The derivative of f(g(x)) is f'(g(x))*g'(x)
• Applications of differentiation include finding maximum and minimum values of a function, determining the concavity of a function, and solving related rates problems.
• Examples of applications: optimization problems (e.g., finding the dimensions of a container with maximum volume), determining whether a company's profit function is increasing or decreasing, determining velocity and acceleration problems

Matrices

• A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
• Matrices are used to represent linear transformations and systems of linear equations.
• Basic matrix operations include:
  • Addition and subtraction: Matrices can be added or subtracted if they have the same dimensions. Corresponding entries are added or subtracted.
  • Scalar multiplication: A matrix can be multiplied by a scalar (single number). Each element in the matrix is multiplied by the scalar.
  • Matrix multiplication: Matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.
  • Matrix transpose: The rows and columns of a matrix are interchanged.
• Special types of matrices include:
  • Identity matrix: A square matrix with 1s on the main diagonal and 0s elsewhere.
  • Zero matrix: A matrix with all entries equal to 0.
  • Square matrix: A matrix with the same number of rows and columns.
  • Row matrix: A matrix with only one row.
  • Column matrix: A matrix with only one column.
• Determinant of a matrix: A scalar value associated with a square matrix; used in various applications e.g., solving systems of linear equations (Cramer's rule).
• Inverse of a matrix: A matrix that, when multiplied by the original matrix, results in the identity matrix. Not all matrices have inverses.
• Applications of matrices include: Solving systems of linear equations, representing linear transformations (geometry), engineering (e.g., structural analysis), image processing, cryptography, economics.