The Product Rule In Calculus: Deriving The Formula And Its Applications.

Product Rule

d/dx [f(x)*g(x)] = f(x)g'(x) + f'(x)g(x)

The Product Rule is a fundamental concept in calculus that is used to find the derivative of a product of two differentiable functions. It states that the derivative of the product of two functions is equal to the sum of the product of the first function with the derivative of the second function and the product of the second function with the derivative of the first function. This can be represented mathematically as follows:

If f(x) and g(x) are two differentiable functions, then

(fg)'(x) = f'(x)g(x) + f(x)g'(x)

where (fg)'(x) is the derivative of the product f(x)g(x), f'(x) and g'(x) are the derivatives of f(x) and g(x) respectively.

The product rule can be used to find the derivative of a wide range of functions, including exponential functions, trigonometric functions, and logarithmic functions. The key is to identify which function is the first function and which is the second function, and then apply the rule accordingly.

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