Parametric Differentiation

Summary When x and y are expressed in terms of a third variable it is called a parameter. To differentiate parametric equations to find derivative of y with respect to x, we use the chain rule method. When the parameter in the equations is “t”, the chain rule is defined as:     What is parameter? We … Read more

The Second Derivative

Summary Second derivative is when we differentiate . We write second derivative of y with respect to x  as:  Stationary Points are obtained by solving the equation  . Maximum point:  Minimum point:  Second derivative Differentiation is the process that we use to find the gradient of a point on the curve. We define a gradient … Read more

Exponentials & Logarithms

Summary is known as an exponential function Logarithmic functions are written as Laws of logarithm: 1.     2.     3.     4.     5.     6.     Exponential functions When n is a positive integer and a is the base in , then n is known as the index, power or exponent. Suppose we have , this is called an exponential … Read more

Integration Techniques

Summary Integration techniques include: Integration of trigonometric functions:  Integration of exponential functions:  Integration of :   Integrating Fractions: This involves , numerator is the derivative of a function within the denominator and lastly partial fractions. We know that the process of antidifferentiation is called integration. Clearly if    then    where    is known as the … Read more

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