How to derive the quotient rule using the limit definition of derivative. скачать в хорошем качестве

How to derive the quotient rule using the limit definition of derivative.

In this video, we continue to develop the algebra of differentiation by considering the derivative of a quotient of two functions. Once we derive the quotient rule, we apply the quotient rule to find the derivative of a rational function (3x^2+1)/(2x-3). To start, we derive the quotient rule using the limit definition of the derivative: limit as h approaches zero Q(x+h)-Q(x) over h, where Q is the quotient of two functions f(x)/g(x). We arrive at a complex fraction in the limit, so we combine the two fractions by finding a common denominator. Next, we have to invent a new term for the numerator, with the goal of finding the expression f(x+h)-f(x) after we factor: this will allow us to find the derivative of f when we take the limit. Subtracting and then adding f(x)g(x) in the numerator gives us two major pieces, where we can factor to find the expression f(x+h)-f(x) and g(x+h)-g(x). Using the limit laws, we rework the limit to highlight the definition of the derivative of f(x) and g(x), simplify and obtain the quotient rule (f'(x)g(x)-f(x)g'(x))/(g(x))^2. After we prove the quotient rule, we apply the quotient rule to taking the derivative of a rational function: we differentiate (3x^2+1)/(2x-3).