Product Rule
If h(x) = f(x)g(x), then h'(x) = f'(x)g(x)+g'(x)f(x)
The product rule is a differentiation formula used to find the derivative of two or more functions that are multiplied together. It states that if u(x) and v(x) are two functions of x, then the derivative of their product is given by:
(d/dx) [u(x) * v(x)] = u(x) * v'(x) + v(x) * u'(x)
Where u'(x) and v'(x) are the derivatives of u(x) and v(x) with respect to x, respectively.
In simpler terms, the product rule tells us that to find the derivative of a product of two functions, we need to differentiate each of the functions separately and then add them together.
For example, suppose we have the function f(x) = x^2 * cos(x). To find its derivative, we would use the product rule as follows:
f'(x) = (d/dx) [x^2 * cos(x)]
= x^2 * (-sin(x)) + cos(x) * 2x
= -x^2 * sin(x) + 2x * cos(x)
Therefore, the derivative of f(x) is f'(x) = -x^2 * sin(x) + 2x * cos(x).
The product rule is a fundamental rule of differential calculus and is used extensively in various applications, such as optimization, physics, and engineering.
More Answers:
Understanding the Cosine Function: Definition, Properties, and ApplicationsUnderstanding Sine(x): Definition, Calculation, and Applications in Trigonometry
Master the Quotient Rule in Calculus: Simplify Derivative Calculations of Quotient of Functions.