Differentiation

Differentiation

Basic Definitions

  • Derivative: The derivative of a function at a point is the rate at which it is changing at that point, represented mathematically as the slope of the function at that point.
  • Differentiation: The process of finding a derivative is called differentiation. Differentiation is a fundamental tool in calculus which allows us to transform a function into its derivative.

Rules of Differentiation

  • Power rule: Differentiation of any variable ‘x’ raised to a power ‘n’ (constant), can be done using the power rule, d/dx[x^n] = n * x^(n-1).
  • Product rule: The derivative of a product of two functions is given by the rule: first function times the derivative of the second function, plus the second function times the derivative of the first function.
  • Quotient rule: The derivative of a quotient of two functions is found by taking the bottom function times the derivative of the top function, minus the top function times the derivative of the bottom function, all over the bottom function squared.
  • Chain rule: This rule is used when differentiating the composition of functions. It states that the derivative of a composite function is the derivative of the outside function evaluated at the inside function times the derivative of the inside function.

Differentiation of Trigonometric Functions

  • The derivative of sin(x) is cos(x).
  • The derivative of cos(x) is -sin(x).
  • The derivative of tan(x) is sec²(x).

Higher Derivatives

  • Higher derivatives are simply derivatives taken multiple times. The second derivative of a function is just the derivative of the derivative of the function.
  • The notation for higher derivatives when we are using ‘f’ notation is to write f’‘(x) for the second derivative and f’’‘(x) for the third derivative. When we have a fourth or higher derivative, we write brackets around the number, as in f^(4)(x), f^(5)(x), etc.

Applications of Differentiation

  • Differentiation is used in a wide variety of fields, such as physics, engineering, economics, and statistics.
  • Some applications include optimizing functions (finding maximum and minimum points), describing motion in terms of velocity and acceleration, and determining rates of change.
  • It can also assist in sketching curves by finding gradient functions and turning points of the curve.

Implicit Differentiation

  • Implicit differentiation is a method for finding derivatives when the function is not expressed explicitly in terms of one variable.
  • Using this technique, one treats the independent variable and dependent variable as equals with respect to differentiation.

Logarithmic Differentiation

  • Logarithmic differentiation is a method used to differentiate functions by taking natural logarithms (ln) of both sides of an equation. This simplifies the process where functions are raised to various powers or when you have a product of functions.

Differentiation of Exponential Functions

  • The exponential functions are unique in a way that the derivative of an exponential function with base ‘e’ (e^x) is itself.
  • The derivative of a^x (where a is a constant base other than e) is a^x * ln(a).