### Video Transcript

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

Here weβve been given a multivariable function. Thatβs a function defined by two or more variables. Here, those variables are π₯ and π¦. Sometimes when dealing with these functions, we want to see what happens when we change just one of the variables. This is called finding the first partial derivative of the function. We use curly dβs or πβs to represent the first partial derivative. And so the first partial derivative with respect to π₯ is ππ by ππ₯.

And so what we do is we imagine that all the other variables, thatβs the variables that are not π₯, are simply constants. Weβre going to treat the variable π¦ as if itβs a constant. And weβre going to differentiate some constant times π to the power of negative π₯ with respect to π₯. And so we recall how we differentiate an expression of the form ππ to the power of ππ₯ with respect to π₯ for real constants π and π.

We simply multiply everything by the coefficient of π₯, and everything else remains unchanged. And we get ππ times π to the power of ππ₯. The coefficient of π₯ in this case is negative one. So ππ by ππ₯ is π¦ squared times negative one times π to the power of negative π₯. Simplifying, and we find our first partial derivative with respect to π₯ is negative π¦ squared times π to the power of negative π₯.