Differentiation

Differentiation

Basic Principles

  • Differentiation is a fundamental concept in calculus that involves finding the derivative of a function.
  • The derivative gives the rate of change of a function at a specific point. This can be used to determine the instantaneous speed, rate of error change, or gradient of a tangent to a function at any point.
  • The process of differentiation consists of finding the limit of the difference quotient.
  • The derivative of a function can be represented as f’(x) or dy/dx.

Rules of Differentiation

  • Power Rule: For any real number n, the derivative of x^n is nx^(n-1).
  • Constant Rule: The derivative of a constant is zero because constants do not change.
  • Sum/Difference Rule: The derivative of the sum or difference of functions is the sum or difference of their derivatives.
  • Product Rule: If you have two functions multiplied together, the derivative is the first function times the derivative of the second, plus the second function times the derivative of the first.
  • Quick Rule or Chain Rule: If a function is composed of two functions, the derivative is the derivative of the outside function times the derivative of the inside function.

Applications of Differentiation

  • Tangents and Normals: Differentiation can be used to find the equation of a tangent to a curve at a given point or the equation of the normal to the curve.
  • Optimization problems: Differentiation lets us find maximum and minimum values of a function which can be used in various real-world optimisation problems.
  • Rate of change: Differentiation can be a way to predict the rate of change in a physical situation where inputs and outputs continuously change.

Examples for Common Differentiation

  • The derivative of y = 5 (a constant function) is 0.
  • The derivative of a linear function, for example y = 3x + 7, is 3, the coefficient of x.
  • If y = x^2, its derivative using the power rule is 2x.
  • Using the sum/difference rule, the derivative of y = 3x^2 + 4x is 6x + 4.
  • For y = x^3 - 5x, its derivative using the power rule and sum/difference rule is 3x^2 - 5.
  • Trigonometric Functions: For instance, the derivative of sin(x) is cos(x), the derivative of cos(x) is -sin(x).

Differentiation is a topic that requires practise to fully grasp. Work through plenty of practise problems to get comfortable with the process.