Video Transcript
By considering the product rule,
find the function π so that π prime of π₯ equals π to the π₯ power over the
square root of π₯ plus two times π to the π₯ power times the square root of π₯.
Weβll first need to remember the
product rule for derivatives. That product rule tells us the
derivative of the function π of π₯ times the function π of π₯ equals π of π₯
times the derivative of π of π₯ plus π of π₯ times the derivative of π of π₯. Before we try to find an π of π₯
and a π of π₯, letβs rewrite this function. We have π prime of π₯ equals π to
the π₯ power. And we know that itβs being
multiplied by one over the square root of π₯. We can write that as π₯ to the
negative one-half power. Weβre multiplying π to the π₯
power times π₯ to the negative one-half power plus two times π to the π₯ power
times π₯ to the one-half power.
Something that we know is that the
derivative of π to the π₯ power equals π to the π₯ power. If we say that π of π₯ equals π
to the π₯ power, then π prime of π₯ also equals π to the π₯ power. This means that π₯ to the negative
one-half power equals π prime of π₯. And it means that π of π₯ equals
two times π₯ to the one-half power. π 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 π of π₯ times π of π₯. And that means the antiderivative
is going to be π of π₯ times π of π₯. We know π of π₯ and we know π of
π₯, which means the antiderivative equals two times π₯ to the one-half power times
π 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 π to the π₯ power.
