The Derivative of tan(x)

In this page we'll talk about the derivative of tan(x). This function is the third most used trigonometric function, but its derivative is less known than the derivatives of sin(x) and cos(x), for example. Here is the formula of its derivative

Proof

To prove this formula, we first express tan(x) as the quotient between sin(x) and cos(x). 

We observe that the expression above is a quotient. So, to calculate the derivative, we can apply the quotient rule. As I explain on that page on the quotient, I prefer applying the product rule in this case.

This is because it is easier for me to recall the product rule. If I had to apply the quotient rule, I would need to look on the internet or a table of formulas to recall what the rule says. So, let's apply my method. 

First, we express the quotient as a product

Now we can apply the product rule

Calculating the derivative of v takes more work, so we'll do that separately. We apply the chain rule

Now we can replace that in the product formula to obtain the derivative of tan(x)

Now we can work on that expression to arrive at the formula shown above

And that's is. This is an excellent example of how to apply my method of calculating derivatives of quotients for forgetful people. If you have a doubt or want to talk about a related problem, just leave me a comment below.

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