### Video Transcript

Find the first partial derivative
with respect to π₯ of the function π of π₯, π¦ equals π to the power of π₯π¦ plus
π₯π¦.

Weβve been given a multivariable
function; itβs a function that involves two variables. Here those are π₯ and π¦. Weβre being asked to find its first
partial derivative. So, what does that mean? We represent this using curly dβs
or π. And we say this is ππ ππ₯. And when we find the first partial
derivative with respect to π₯, essentially we differentiate as normal but treat all
the other variables as if they are constants.

So, this case, weβre going to hold
the π¦βs constant, and weβre going to differentiate π to the power of π₯π¦ plus
π₯π¦ treating the π¦βs as constants. So, letβs begin by recalling how we
differentiate π to the power of ππ₯ for some constant π. We get π times π to the power of
ππ₯. And so, the first partial
derivative with respect to π₯ of π to the power of π₯π¦ will be π¦ times π to the
power of π₯π¦.

Similarly, when we differentiate an
expression of the form ππ₯ for real constants π with respect to π₯, we simply get
π. And so, the first partial
derivative with respect to π₯ of π₯π¦ must simply be π¦. We could leave it in this form. Or we can factor by taking out a
constant factor of π¦. When we do, we find the first
partial derivative with respect to π₯ of our function, which weβve written ππ
ππ₯, is equal to π¦ times π to the power of π₯π¦ plus one.