Video: Using the Product Rule

By considering the product rule, find the function 𝑓 so that 𝑓′(π‘₯) = (𝑒^(π‘₯)/√π‘₯) + 2𝑒^(π‘₯) √π‘₯.

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Video Transcript

By considering the product rule, find the function 𝑓 so that 𝑓 prime of π‘₯ equals 𝑒 to the π‘₯ power over the square root of π‘₯ plus two times 𝑒 to the π‘₯ power times the square root of π‘₯.

We’ll first need to remember the product rule for derivatives. That product rule tells us the derivative of the function 𝑓 of π‘₯ times the function 𝑔 of π‘₯ equals 𝑓 of π‘₯ times the derivative of 𝑔 of π‘₯ plus 𝑔 of π‘₯ times the derivative of 𝑓 of π‘₯. Before we try to find an 𝑓 of π‘₯ and a 𝑔 of π‘₯, let’s rewrite this function. We have 𝑓 prime of π‘₯ equals 𝑒 to the π‘₯ power. And we know that it’s being multiplied by one over the square root of π‘₯. We can write that as π‘₯ to the negative one-half power. We’re multiplying 𝑒 to the π‘₯ power times π‘₯ to the negative one-half power plus two times 𝑒 to the π‘₯ power times π‘₯ to the one-half power.

Something that we know is that the derivative of 𝑒 to the π‘₯ power equals 𝑒 to the π‘₯ power. If we say that 𝑓 of π‘₯ equals 𝑒 to the π‘₯ power, then 𝑓 prime of π‘₯ also equals 𝑒 to the π‘₯ power. This means that π‘₯ to the negative one-half power equals 𝑔 prime of π‘₯. And it means that 𝑔 of π‘₯ equals two times π‘₯ to the one-half power. 𝑔 of π‘₯ equals two times π‘₯ to the one-half power. If we check that derivative, we get two times one-half times π‘₯ to the one-half minus one power, which is in fact π‘₯ to the negative one-half power. But what does this mean for us? Well, in the product rule, this value is the derivative of 𝑓 of π‘₯ times 𝑔 of π‘₯. And that means the antiderivative is going to be 𝑓 of π‘₯ times 𝑔 of π‘₯. We know 𝑓 of π‘₯ and we know 𝑔 of π‘₯, which means the antiderivative equals two times π‘₯ to the one-half power times 𝑒 to the π‘₯ power. And we can put that back in the form it was given to us in. Two times the square root of π‘₯ times 𝑒 to the π‘₯ power.

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