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.
