# Video: Using the Product Rule

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

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### Video Transcript

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

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

Something that we know is that the derivative of 𝑒 to the 𝑥 power equals 𝑒 to the 𝑥 power. If we say that 𝑓 of 𝑥 equals 𝑒 to the 𝑥 power, then 𝑓 prime of 𝑥 also equals 𝑒 to the 𝑥 power. This means that 𝑥 to the negative one-half power equals 𝑔 prime of 𝑥. And it means that 𝑔 of 𝑥 equals two times 𝑥 to the one-half power. 𝑔 of 𝑥 equals two times 𝑥 to the one-half power. If we check that derivative, we get two times one-half times 𝑥 to the one-half minus one power, which is in fact 𝑥 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 𝑥 times 𝑔 of 𝑥. And that means the antiderivative is going to be 𝑓 of 𝑥 times 𝑔 of 𝑥. We know 𝑓 of 𝑥 and we know 𝑔 of 𝑥, which means the antiderivative equals two times 𝑥 to the one-half power times 𝑒 to the 𝑥 power. And we can put that back in the form it was given to us in. Two times the square root of 𝑥 times 𝑒 to the 𝑥 power.