Using Khan Academy Product Rule Calculus

In calculus, the product rule is a fundamental concept used to find the derivative of a function that is the product of two or more other functions. In this article, we will explore how the product rule can be used in conjunction with the quotient rule to solve problems involving rational expressions.

To begin, let's consider a simple example. Suppose we have a function y = x/(x^2 + 3x). To find the derivative of this function, we can use the quotient rule, which states that if y = u/v, then dy/dx = (v(du/dx) - u(dv/dx))/v^2.

In our example, u(x) = x and v(x) = x^2 + 3x. Using the product rule, we can find the derivatives of both u(x) and v(x). The derivative of u(x) is simply du/dx = 1, while the derivative of v(x) is dv/dx = 2x + 3.

Now, substituting these values into the quotient rule formula, we get:

dy/dx = ((x^2 + 3x)(1) - x(2x + 3))/((x^2 + 3x)^2)

Simplifying this expression, we get:

dy/dx = (x^2 + 3x - 2x^2 - 3x)/(x^4 + 6x^3 + 9x^2)

This is the derivative of our original function y(x) = x/(x^2 + 3x). We can simplify this expression further by factoring the numerator and combining like terms:

dy/dx = (-1x^2 - 3x)/(x^4 + 6x^3 + 9x^2)

This is the final answer. Note that the quotient rule allows us to find the derivative of a rational function without having to use logarithmic differentiation or other advanced techniques.

In conclusion, the product rule and quotient rule are powerful tools in calculus that can be used to solve problems involving rational expressions. By understanding how these rules work, you can develop a deeper appreciation for the beauty and elegance of calculus.

Kahn academy product rule calculus