# What is the derivative of #tan(x+y) = ln x + 5y#?

##### 1 Answer

#### Explanation:

Since you have an equation mixed with **implicit differentiation.** If you're not sure what that really is/does, I'd recommend watching my video on it:

Anyways, the 3 key steps to implicit differentiation are:

- Differentiate terms with an
#x# in them as usual. - Differentiate terms with a
#y# in them as normal.- BUT tag on a
#dy/dx# to whatever you get.

- BUT tag on a
- Solve for
#dy/dx# .

So, with this in mind, we start by taking the derivative of both sides of this equation (with respect to

I'm not going to walk through the intricacies of actually doing the derivatives, since this is not a focus of this problem, but know that you will need a chain rule to evaluate the derivative of the left hand side of that equation.

You should get:

Now, I'm going to distribute my

The reason I did this is just so that I have two single terms with a

Now, we factor out the

...and that is a perfectly good final answer. However, fractions in fractions aren't nice to look at, so we're going to multiply through by

This would be the final answer that you'd most likely see in your textbook, or in other literary sources. Note that it's perfectly OK to leave your answer in terms of

Hope that helped :)