Differentiation

Differentiation

  • Differentiation is a mathematical procedure involving calculus to find the rate of change at a certain point.

  • Differentiation can be applied to a range of mathematical problems including finding minimum or maximum values in problems, determining rates of change or finding tangents to curves.

Basic Concept

  • The derivative of a function measures the rate of change of the output value with respect to its input.

  • The derivative of a constant is zero since constants do not change.

  • Power rule: The derivative of x^n, where n is any real number, is n*x^(n-1).

  • The derivative of a function that is the sum or difference of two functions is simply the sum or difference of their derivatives.

Product and Quotient Rules

  • Product rule: The derivative of the product of two functions is the derivative of the first times the second plus the first times the derivative of the second.

  • Quotient rule: The derivative of the quotient of two functions is the bottom times the derivative of the top minus the top times the derivative of the bottom, all over the squared of the bottom function.

Chain Rule

  • The Chain Rule allows us to differentiate composite functions. This is when a function is composed of several other functions.

Applications of Differentiation

  • Differentiation is used to find maximum and minimum values of a function. A function reaches its minimum or maximum where its derivative is zero.

  • Differentiation can be used to solve problems of optimisation, such as finding the minimum cost or maximum profit.

  • Differentiation is applied to find tangents to curves at given points. The derivative of a function at a certain point gives the slope of the tangent to the graph at this point.

Higher Order Derivatives

  • The derivative of a function can be differentiated again to get the second derivative, and this process can be continued to get higher order derivatives.

  • The second derivative provides information about the curvature or concavity of the graph of the function.

Remember, practising differentiation on a range of different function types is the key part of mastering this topic.