Product Rule: d/dx [ f(x) * g(x) ]
f'(x)g(x) + f(x)g'(x)
The product rule is a differentiation rule that tells us how to take the derivative of a function that is the product of two other functions. The product rule can be stated as follows:
If f(x) and g(x) are two differentiable functions, then the derivative of their product f(x) * g(x) is given by the formula:
d/dx [ f(x) * g(x) ] = f'(x) * g(x) + f(x) * g'(x)
where f'(x) and g'(x) represent the derivatives of f(x) and g(x), respectively.
In other words, to find the derivative of a product of two functions, we:
1. Take the derivative of the first function and multiply it by the second function.
2. Add to that the derivative of the second function multiplied by the first function.
For example, let’s say we have the function f(x) = x^2 * sin(x), and we want to find its derivative.
Using the product rule, we have:
f'(x) = d/dx [ x^2 * sin(x) ]
= (d/dx [ x^2 ] * sin(x)) + (x^2 * d/dx [ sin(x) ])
= (2x * sin(x)) + (x^2 * cos(x))
Therefore, the derivative of f(x) = x^2 * sin(x) is f'(x) = (2x * sin(x)) + (x^2 * cos(x)).
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