Video: Finding the First Partial Derivative of a Multivariable Function of Two Variables Involving Exponential Functions

Video Transcript

Find the first partial derivative with respect to 𝑥 of the function 𝑓 of 𝑥, 𝑦 equals 𝑒 to the power of 𝑥𝑦 plus 𝑥𝑦.

We’ve been given a multivariable function; it’s a function that involves two variables. Here those are 𝑥 and 𝑦. We’re being asked to find its first partial derivative. So, what does that mean? We represent this using curly d’s or 𝜕. And we say this is 𝜕𝑓 𝜕𝑥. And when we find the first partial derivative with respect to 𝑥, essentially we differentiate as normal but treat all the other variables as if they are constants.

So, this case, we’re going to hold the 𝑦’s constant, and we’re going to differentiate 𝑒 to the power of 𝑥𝑦 plus 𝑥𝑦 treating the 𝑦’s as constants. So, let’s begin by recalling how we differentiate 𝑒 to the power of 𝑎𝑥 for some constant 𝑎. We get 𝑎 times 𝑒 to the power of 𝑎𝑥. And so, the first partial derivative with respect to 𝑥 of 𝑒 to the power of 𝑥𝑦 will be 𝑦 times 𝑒 to the power of 𝑥𝑦.

Similarly, when we differentiate an expression of the form 𝑎𝑥 for real constants 𝑎 with respect to 𝑥, we simply get 𝑎. And so, the first partial derivative with respect to 𝑥 of 𝑥𝑦 must simply be 𝑦. We could leave it in this form. Or we can factor by taking out a constant factor of 𝑦. When we do, we find the first partial derivative with respect to 𝑥 of our function, which we’ve written 𝜕𝑓 𝜕𝑥, is equal to 𝑦 times 𝑒 to the power of 𝑥𝑦 plus one.