derivative of product
To find the derivative of a product of two functions, we can use the product rule
To find the derivative of a product of two functions, we can use the product rule. The product rule states that if we have two functions, f(x) and g(x), then the derivative of their product, f(x) * g(x), 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, you keep one of the functions the same and differentiate the other, and then vice versa, and finally add the two results together.
Let’s go through an example to illustrate this.
Example: Find the derivative of the product of two functions, f(x) = x^2 and g(x) = 3x.
Step 1: Identify the functions f(x) and g(x).
In this example, f(x) = x^2 and g(x) = 3x.
Step 2: Differentiate f(x) and g(x) separately.
f'(x) = 2x (Applying the power rule, where d/dx(x^n) = n*x^(n-1))
g'(x) = 3 (The derivative of 3x with respect to x is simply 3, as the derivative of a constant is always zero)
Step 3: Apply the product rule.
(f(x) * g(x))’ = f'(x) * g(x) + f(x) * g'(x)
(f(x) * g(x))’ = (2x * 3x) + (x^2 * 3)
Simplifying the above equation gives us the derivative of f(x) * g(x):
(f(x) * g(x))’ = 6x^2 + 3x^2
So, the derivative of the product of f(x) = x^2 and g(x) = 3x is (6x^2 + 3x^2).
I hope this explanation helps you understand how to take the derivative of a product using the product rule. Let me know if you have any more questions!
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