Question Video: Finding the First Partial Derivative of a Two-Variable Function

Video Transcript

Find the first partial derivative with respect to 𝑦 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. Here, that’s 𝑥 and 𝑦. We’re looking to find the first partial derivative with respect to 𝑦 of our function. And we denote that as shown. Now, the reason we use this curly ∂ is because we’re looking to see how our function changes as we let just one of the variables change. In this case, that’s 𝑦. And we hold all the others constant. So we change the variable 𝑦 and we treat 𝑥 as a constant here.

So let’s go term by term. The first term in our function is 𝑥. Remember, we’re finding the first partial derivative with respect to 𝑦. So we treat 𝑥 as a constant. And we know when we differentiate a constant with respect to 𝑦, we get zero. Then, the second term is two 𝑦. We know that when we differentiate two 𝑦 with respect to 𝑦, we get two. And zero plus two is two.

The first partial derivative with respect to 𝑦 of the function 𝑓 of 𝑥, 𝑦 equals 𝑥 plus two 𝑦 is two.