Product rule
f’g + g’f
In calculus, the product rule is a formula used to find the derivative of the product of two functions. The product rule states that if we have two functions f(x) and g(x), then the derivative of their product fg(x) is:
(fg)’ = f’g + fg’
In other words, to find the derivative of the product of two functions, we 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.
For example, let’s say we have f(x) = x^2 and g(x) = sin(x). The product of these two functions is:
fg(x) = x^2 * sin(x)
To find the derivative of fg(x), we use the product rule:
(fg)’ = f’g + fg’
f'(x) = 2x (using the power rule for differentiation)
g'(x) = cos(x) (using the differentiation rule for sine function)
So, (fg)’ = (2x * sin(x)) + (x^2 * cos(x))
Therefore, the derivative of the product of f(x) and g(x) is:
fg(x)’ = (2x * sin(x)) + (x^2 * cos(x))
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