Find the first partial derivative with respect to 𝑥 of the function 𝑓 of 𝑥, 𝑦 equals 𝑦 squared times 𝑒 to the power of negative 𝑥.
Here we’ve been given a multivariable function. That’s a function defined by two or more variables. Here, those variables are 𝑥 and 𝑦. 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 d’s or 𝜕’s to represent the first partial derivative. And so the first partial derivative with respect to 𝑥 is 𝜕𝑓 by 𝜕𝑥.
And so what we do is we imagine that all the other variables, that’s the variables that are not 𝑥, are simply constants. We’re going to treat the variable 𝑦 as if it’s a constant. And we’re going to differentiate some constant times 𝑒 to the power of negative 𝑥 with respect to 𝑥. And so we recall how we differentiate an expression of the form 𝑎𝑒 to the power of 𝑏𝑥 with respect to 𝑥 for real constants 𝑎 and 𝑏.
We simply multiply everything by the coefficient of 𝑥, and everything else remains unchanged. And we get 𝑎𝑏 times 𝑒 to the power of 𝑏𝑥. The coefficient of 𝑥 in this case is negative one. So 𝜕𝑓 by 𝜕𝑥 is 𝑦 squared times negative one times 𝑒 to the power of negative 𝑥. Simplifying, and we find our first partial derivative with respect to 𝑥 is negative 𝑦 squared times 𝑒 to the power of negative 𝑥.