Find the first partial derivative with respect to 𝑥 of the function 𝑓 of 𝑥, 𝑦 equals 𝑥 plus two 𝑦.
This is a multivariable function, a function in terms of more than one variable. So here, that’s 𝑥 and 𝑦. We’re being asked to find the first partial derivative with respect to 𝑥 of our function. So we denote that as shown using these sort of swirly d’s.
Essentially, when we’re finding the first partial derivative with respect to 𝑥, we’re interested in saying how the function changes as we let just one of the variables change, as we let 𝑥 change. So we hold all the other variables constant. So let’s see what that might look like. And we’re going to treat this term by term.
The derivative of 𝑥 with respect to 𝑥 is just one. Now, remember, we’re holding 𝑦 constant. So we differentiate as if 𝑦 is a constant. And then the derivative of two 𝑦 with respect to 𝑥 must be zero. And that means the first partial derivative with respect to 𝑥 of our function 𝑓 of 𝑥, 𝑦 is one.