# How do you differentiate #y=csc^-1x-4cot^-1x#?

##### 1 Answer

#### Explanation:

Actually it isn't too bad!

**Step I**

Let

I shall assume that there is no need to show the step by step differentiation of

We see that

**Step II**

Let

Again, if I may, I shall skip right to:

Again, using the same Pythagorean Identity, we have:

**Step III**

Now combine the lot:

*Note: remember to re-introduce the constant #4# which I left out whilst differentiating #4cot^-1x#*

*Second note: the absolute value bars rather mysteriously appeared on the denominator of that derivative. I believe this is because actually sketching the graph of #y=csc^-1x#, the function is strictly decreasing, so by placing absolute value bars around the stray #x#, ensures that the derivative function stays negative for all #x#.*

*Third note: the domain comes in because the #csc# function is defined across this domain. The #cot# function is defined for all #x#, so the combined function takes the domain of the former.*