# Video: Using the Product Rule

Suppose that π(2) = 3, π(2) = 5, πβ²(2) = β1, and πβ²(2) = 6. Evaluate (π(π₯)π(π₯))β² β πβ²(π₯)πβ²(π₯) at π₯ = 2.

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### Video Transcript

Suppose that π of two equals three, π of two equals five, π prime of two equals negative one, and π prime of two equals six. Evaluate π of π₯ π of π₯ prime, whereas π prime of π₯ π prime of π₯ at π₯ equals two.

Weβve been given the values of two functions π of π₯ and π of π₯ and their derivatives at π₯ equals two. And weβre asked to evaluate this more complicated expression involving derivatives. We are able to substitute some values straightaway. Weβre told in the question that π prime of two is equal to negative one. So when evaluating this expression at π₯ equals two, weβll be able to swap π prime of π₯ for negative one. Weβre also told that π prime of two is equal to six. So when evaluating this expression π₯ equals two, weβll be able to swap π prime of π₯ for six.

What we donβt know is the value of π of π₯ π of π₯ all prime when π₯ is equal to two. Now, this means we find the product of the functions π of π₯ and π of π₯ and then differentiate it. So we need to recall how to find the derivative of a product of functions. We need to use the product rule, which tells us that for two differentiable functions π of π₯ and π of π₯ the derivative of their product π of π₯ π of π₯ is equal to π times π prime plus π times π prime. We multiply each function by the derivative of the other and add these together.

So we now know how to find π of π₯ π of π₯ all prime in terms of π of π₯, π of π₯, and their derivatives. To evaluate this derivative at π₯ equals two, we just need to substitute two for π₯. So we have that π of two π of two all prime is equal to π of two multiplied by π prime of two plus π of two multiplied by π prime of two. And each of these values is given in the question. Remember, π prime of two is equal to negative one and π prime of two is equal to six. Weβre also given that π of two is equal to three and π of two is equal to five.

So we have all of the values we need to evaluate this. So that full expression π of π₯ π of π₯ prime minus π prime of π₯ π prime of π₯ all evaluated at π₯ equals two is equal to three times six plus five times negative one minus negative one times six. Thatβs 18 minus five minus negative six, which is equal to 19. So by recalling the product rule, which tells us how to find the derivative of a product of two differentiable functions and then substituting the relevant values of π of two, π of two, π prime of two, and π prime of two. We found that the value of this expression when π₯ is two is 19.