Product Rule
The product rule is a useful tool in calculus for finding the derivative of a product of two functions
The product rule is a useful tool in calculus for finding the derivative of a product of two functions. It states that if you have two functions, let’s call them f(x) and g(x), then the derivative of their product is given by:
(f(x) * g(x))’ = f'(x) * g(x) + f(x) * g'(x)
In other words, to find the derivative of a product of two functions, you have to take the derivative of the first function and multiply it by the second function, then add it to the product of the first function and the derivative of the second function.
To better understand the product rule, let’s look at an example:
Suppose we have the functions f(x) = 2x^3 and g(x) = 5x^2. We want to find the derivative of their product, f(x) * g(x).
Step 1: First, we need to find the derivative of f(x) and g(x) individually.
f'(x) = 6x^2 (using the power rule, where you bring down the exponent and multiply it by the coefficient)
g'(x) = 10x (again, using the power rule)
Step 2: Now, we can use the product rule to find the derivative of the product of f(x) and g(x).
(f(x) * g(x))’ = f'(x) * g(x) + f(x) * g'(x)
(2x^3 * 5x^2)’ = 6x^2 * 5x^2 + 2x^3 * 10x
Step 3: Simplify the expression.
(2x^3 * 5x^2)’ = 30x^4 + 20x^4
(2x^3 * 5x^2)’ = 50x^4
So, the derivative of the product f(x) * g(x) is 50x^4.
Remember to always apply the product rule when you have to differentiate a product of two functions. It is an essential tool in calculus.
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