The Product Rule In Calculus: How To Find The Derivative Of A Product Of Two Functions

Product rule of f(x)g(x)

f'(x)g(x)+g'(x)f(x)

The product rule is a fundamental rule in calculus that allows us to find the derivative of a product of two functions. If we have two functions f(x) and g(x), then the product rule tells us that the derivative of their product f(x)*g(x) with respect to x is:

(f(x)g(x))’ = f'(x)g(x) + f(x)g'(x)

This means that we take the derivative of the first function, f(x), and multiply it by the second function, g(x). Then we add to this the derivative of the second function, g(x), multiplied by the first function, f(x).

To see why this rule works, let’s expand the expression for the derivative of the product using the definition of the derivative:

(f(x)g(x))’ = lim(h->0) [(f(x+h)g(x+h) – f(x)g(x))/h]

Expanding the product, we get:

(f(x)g(x))’ = lim(h->0) [(f(x+h)g(x+h) – f(x)g(x) – f(x)g(x+h) + f(x)g(x+h))/h]

Now we can use algebraic manipulation to rewrite this expression as:

(f(x)g(x))’ = lim(h->0) [f(x+h)g(x+h) – f(x)g(x+h) + f(x)g(x+h) – f(x)g(x)])/h]

Using the distributive property, we can write this as:

(f(x)g(x))’ = lim(h->0) [g(x+h)(f(x+h) – f(x)) + f(x)(g(x+h) – g(x))]/h]

Now we can use the definition of the derivative again to rewrite this expression as:

(f(x)g(x))’ = g(x)f'(x) + f(x)g'(x)

which is the product rule.

Therefore, the product rule tells us that the derivative of the product of two functions f(x) and g(x) is equal to the derivative of the first function times the second function plus the derivative of the second function times the first function.

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