Question Video: Finding the First Partial Derivative in a Multivariable Function of Two Variables | Nagwa Question Video: Finding the First Partial Derivative in a Multivariable Function of Two Variables | Nagwa

# Question Video: Finding the First Partial Derivative in a Multivariable Function of Two Variables Mathematics

Find the first partial derivative with respect to π₯ of the function π(π₯, π¦) = π¦Β²π^(βπ₯).

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### 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 weβve been given a multivariable function. Thatβs 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 dβs or πβs to represent the first partial derivative. And so the first partial derivative with respect to π₯ is ππ by ππ₯.

And so what we do is we imagine that all the other variables, thatβs the variables that are not π₯, are simply constants. Weβre going to treat the variable π¦ as if itβs a constant. And weβre 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 ππ to the power of ππ₯ 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 ππ times π to the power of ππ₯. The coefficient of π₯ in this case is negative one. So ππ by ππ₯ 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₯.