if 𝑓 of 𝑥 is equal to four 𝑥
squared plus six plus two ln four 𝑥, find 𝑓 prime of two.
𝑓 prime is the derivative of our
function with respect to 𝑥. So we’re going to need to
differentiate four 𝑥 squared plus six plus two ln four 𝑥. Let’s do this by individually
differentiating four 𝑥 squared, six, and two ln four 𝑥 with respect to 𝑥. Recall the derivative of 𝑎𝑥 to
the power of 𝑛, where 𝑛 is a real number and 𝑎 is a constant, is 𝑛𝑎𝑥 to the
power of 𝑛 minus one. And this means we can differentiate
four 𝑥 squared with respect to 𝑥. It’s two times four 𝑥 to the power
Well, two times four is eight and
𝑥 to the power of one is just 𝑥. So the derivative of four 𝑥
squared with respect to 𝑥 is eight 𝑥. When we differentiate a constant
such as the six here, we get zero. But what about the derivative of
two ln of four 𝑥? Well, we know that for constant
values of 𝑎, the derivative of ln 𝑎𝑥 with respect to 𝑥 is one over 𝑥. And we use the constant multiple
rule. This says that we are allowed to
take a constant outside of a derivative and concentrate on differentiating the
function of 𝑥 itself.
So the derivative of two ln of four
𝑥 with respect to 𝑥 is two times one over 𝑥 which is two over 𝑥. And so, we can see that 𝑓 prime of
𝑥 here is eight 𝑥 plus zero plus two over 𝑥, which of course we can write as
eight 𝑥 plus two over 𝑥.
Now, we’re not quite finished. We’re looking to find 𝑓 prime of
two. So we substitute 𝑥 is equal to two
into our expression for the derivative. And that gives us eight times two
plus two over two. Eight times two is 16 and two
divided by two is one. 16 plus one is 17.
So if 𝑓 of 𝑥 is equal to four 𝑥
squared plus six plus two ln of four 𝑥, 𝑓 prime of two is 17.