Differential Equations

Summary Differential Equation – any equation which involves or any higher derivative. Solving differential equations means finding a relation between y and x alone through integration. We use the method of separating variables in order to solve linear differential equations. We must be able to form a differential equation from the given information. A differential … Read more

Implicit Differentiation

Summary An implicit equation is an equation which is not in the form , it consists of two variable x and y which cannot be separated. Implicit Functions are differentiated by using ”chain rule” in combination with the ”product and quotient rule”. When we differentiate y we write with the derivative i.e To find the … Read more

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:     We now completely understand … 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:    Differentiation is the process that we use to find the gradient of a point on the curve. We define a gradient function … 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