### Video Transcript

Find the first partial derivative with respect to π¦ of the function π of π₯, π¦, π§ is equal to the natural log of π₯ plus two π¦ plus three π§.

Here, weβve been given a multivariable function. Itβs a function defined by three variables. Those are π₯, π¦, and π§. Weβre looking to find its first partial derivative with respect to π¦. We represent this as ππ by ππ¦. When we find the first partial derivative with respect to π¦, we treat all the other variables as if they were constants and then differentiate with respect to π¦. So, in the expression the natural log of π₯ plus two π¦ plus three π§, weβre going to treat the variables π₯ and π§ as if they are constants.

And so, letβs recall how we differentiate an expression of the form the natural log of ππ₯ plus π with respect to π₯, where π and π are constants. The derivative is π over ππ₯ plus π. Note, changing the variable does not change the derivative. We see that the derivative of the natural log of ππ¦ plus π with respect to π¦ is π over ππ¦ plus π. And so, weβre ready to find the first partial derivative with respect to π¦ of our function.

The coefficient of π¦ becomes the numerator of our expression. So, the coefficient of π¦ is two, and our numerator is two. Then, the denominator is the inner part of our composite function. So, itβs π₯ plus two π¦ plus three π§. And this means the first partial derivative with respect to π¦, thatβs ππ ππ¦, is two over π₯ plus two π¦ plus three π§.