Integration by Substitution

Summary Substitution Rule is defined as: Indefinite Integral: Definite Integral: where   We often get an integral which does not correspond to any standard result mentioned earlier. We therefore choose a suitable new variable u to replace x. We also replace dx by du. If this substitution transforms the original integrand into a simpler integral … Read more

The Product and Quotient Rule

Summary The Product Rule Formula: The Quotient Rule Formula: Where f’(x) and g’(x) are derivatives of f(x) and g(x) respectively.   The Product Rule When we have to find the derivative of the product of two functions, we apply ”The Product Rule”. Let’s suppose we have two functions f(x) and g(x). We are required to … Read more

Tangents and Normals

Summary A tangent is a line that just touches the curve but doesn’t go through it. The gradient of a tangent = Gradient of a curve at that point A normal is a straight line perpendicular (at right angle 90°) to a curve. We know that differentiation is the process that we use to find … Read more

The Chain Rule

Summary Chain rule lets us differentiate a function of a function i.e  Chain rule can be applied through two different equations: AND What is the chain rule? One of the rules of differentiation is the chain rule, which basically lets us differentiate a function of a function in other words differentiation of composite functions. Let’s … Read more

Uses Of Differentiation

Summary If y = f(x) is an increasing function then  If y = f(x) is a decreasing function then  Stationary Points are obtained by solving the equation  3 types of stationary points: i) Maximum Point ii) Minimum Point iii) Point of Inflexion Increasing Function Refer to Fig 1, consider a point A on the curve … Read more

Differentiation From First Principle

Summary 4 steps to work out differentiation from the First Principle: Give increments to both x & y i.e . Find change of y. Find rate of change of y with respect to x i.e    or  . Take the limit of    as . Gradient of a curve We know that the gradient of … Read more