Video Transcript
Find 𝑓 prime of 𝑥 if 𝑓 of 𝑥 is
equal to cot 𝑥 minus csc squared 𝑥.
Here, we have a trigonometric
function in 𝑥 for which we’re looking to find the derivative. That’s 𝑓 prime of 𝑥. We can differentiate each part of
the expression separately. To find the derivative of cot of 𝑥
or the cotangent of 𝑥 with respect to 𝑥, we use the standard result. That says that the derivative of
cot of 𝑥 with respect to 𝑥 is negative csc squared 𝑥.
But what about the derivative of
csc squared of 𝑥? We’re going to rewrite csc squared
of 𝑥 as csc 𝑥 squared. Remember these mean exactly the
same thing. It’s just notation. But by writing like this, we can
see that we have a function of a function. It’s a composite function. And this means we can use the chain
rule to differentiate csc 𝑥 squared.
The chain rule says that if 𝑦 is a
function of 𝑢 and 𝑢 is a function of 𝑥, then the derivative of 𝑦 with respect to
𝑥 is equal to d𝑦 by d𝑢 times d𝑢 by d𝑥. So we let 𝑢 be equal to csc
𝑥. And that means 𝑦 is equal to 𝑢
squared. To use the chain rule, we’re going
to need to differentiate 𝑢 with respect to 𝑥 and differentiate 𝑦 with respect to
𝑢.
Once again, we recall a standard
result. The derivative of csc 𝑥 with
respect to 𝑥 is negative csc 𝑥 cot 𝑥. So our d𝑢 by d𝑥 is negative csc
𝑥 cot 𝑥. The derivative of 𝑢 squared with
respect to 𝑢 is two times 𝑢 to the power of one or just two 𝑢. This means then that the derivative
of csc squared 𝑥 with respect to 𝑥 is negative csc 𝑥 cot 𝑥 times two 𝑢.
But we stated that 𝑢 was equal to
csc 𝑥. So we replace 𝑢 with csc 𝑥. And the derivative of csc squared
𝑥 with respect to 𝑥 is negative two csc squared 𝑥 cot 𝑥. So 𝑓 prime of 𝑥 is negative csc
squared 𝑥 minus negative two csc squared 𝑥 cot 𝑥. Well, a negative multiplied by a
negative is a positive. So we rewrite this a little
bit. And then, we spot that we have a
common factor of csc squared 𝑥.
We fully factor our expression and
we see that 𝑓 prime of 𝑥 the derivative of our function with respect to 𝑥 is csc
squared 𝑥 multiplied by two cot 𝑥 minus one.
