Video Transcript
Differentiate 𝑓 of 𝑥 equals five
times sin of five times the natural log of 𝑥.
This is a composite function. It’s a function of a function. And we can, therefore, use the
chain rule to find its derivative. This says that if 𝑦 is some
function in 𝑢 and 𝑢 is some function in 𝑥, then d𝑦 by d𝑥 is equal to d𝑦 by d𝑢
times d𝑢 by d𝑥. We’re going to let 𝑢 be equal to
five times the natural log of 𝑥. And the derivative of the natural
log of 𝑥 is one over 𝑥. So d𝑢 by d𝑥 is five times it;
it’s five times one over 𝑥, which is simply five over 𝑥. Instead of using 𝑓 of 𝑥, let’s
use 𝑦. And this means that 𝑦 is equal to
five sin of 𝑢. And the derivative of sin of 𝑢 is
cos of 𝑢. So d𝑦 by d𝑢 is five cos 𝑢. And according to the chain rule d𝑦
by d𝑥 is the product of these.
Referring back to the original
notation, we can say that 𝑓 prime of 𝑥, the derivative, is five over 𝑥 times five
cos of 𝑢. We’ll replace 𝑢 with five times
the natural log of 𝑥. And we see that, in terms of 𝑥, 𝑓
prime of 𝑥 is five over 𝑥 times five cos of five times the natural log of 𝑥. And then we simplify and we see
theat 𝑓 prime of 𝑥 is 25 over 𝑥 times the cosine of five times the natural log of
𝑥.