Product Rule
The product rule is a formula used to find the derivative of a function that is the product of two other functions
The product rule is a formula used to find the derivative of a function that is the product of two other functions. It is commonly used in calculus to differentiate functions that are multiplied together.
The product rule states that if we have two functions, f(x) and g(x), their product f(x) * g(x) can be differentiated using the following formula:
(d/dx) [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x)
Here, f'(x) represents the derivative of f(x) with respect to x, and g'(x) represents the derivative of g(x) with respect to x.
To apply the product rule, we need to differentiate each function separately and then combine the results using the formula. Let’s look at an example:
Example: Find the derivative of h(x) = x^2 * sin(x)
To use the product rule, we first need to identify the two functions being multiplied: f(x) = x^2 and g(x) = sin(x).
Next, we find the derivatives of each function:
f'(x) = 2x (derivative of x^2)
g'(x) = cos(x) (derivative of sin(x))
Finally, we apply the product rule:
(d/dx) [x^2 * sin(x)] = f'(x) * g(x) + f(x) * g'(x)
= 2x * sin(x) + x^2 * cos(x)
So, the derivative of h(x) = x^2 * sin(x) is 2x * sin(x) + x^2 * cos(x).
The product rule allows us to differentiate functions that are the product of two others, which can be useful in various areas of mathematics, physics, and engineering where such functions are commonly encountered.
More Answers:
Why the derivative of a constant function is always zero | Explained with mathThe Quotient Rule | How to Find the Derivative of a Quotient of Two Functions
Mastering the Power Rule | Easily Finding Derivatives of Functions Raised to a Power