Using Differentiation

Using Differentiation

Understanding Differentiation

  • Differentiation is a fundamental concept in calculus that allows us to find rates of change and the slope of a curve at a specific point.
  • It involves the use of derivatives, which are mathematical tools that let us measure how one quantity changes in relation to another.
  • When we differentiate a function, we are effectively finding the derivative of that function.

Basic Differentiation Rules

  • The power rule: If you have a function of the form f(x) = x^n, where n is any real number, then the derivative, f’(x) is n*x^(n-1).
  • The derivative of a constant: The derivative of a constant is zero, because a constant does not change.
  • The sum and difference rules: The derivative of the sum or difference of two functions is just the sum or difference of their derivatives.
  • The 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.

Chain Rule

  • The Chain Rule is a special rule used when differentiating compositions of functions.
  • In essence, it involves finding the derivative of the outer function and then multiplying by the derivative of the inner function.

Implicit Differentiation

  • Implicit Differentiation is used when it is difficult to solve the equation for a specific variable before differentiating.
  • This involves differentiating each term of the equation, treating ‘y’ as an implicit function of ‘x’.

Applications of Differentiation

  • Differentiation has several practical applications such as in physics, for understanding motion and forces and in economics, for calculating marginal cost and marginal revenue.
  • It can help determine optimum points – maximum, minimum and points of inflection, which are critical for problem-solving in several fields.

Higher Derivatives

  • The second derivative of a function gives us the rate at which the first derivative is changing, also known as the concavity of the function.
  • Concativity can provide important information about the behaviour of a function, for example indicating whether a point is a maximum, a minimum, or a point of inflection.