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

Find the first partial derivative with respect to π‘₯ of the function 𝑓(π‘₯, 𝑦) = π‘₯ + 2𝑦.

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Video Transcript

Find the first partial derivative with respect to π‘₯ of the function 𝑓 of π‘₯, 𝑦 equals π‘₯ plus two 𝑦.

This is a multivariable function, a function in terms of more than one variable. So here, that’s π‘₯ and 𝑦. We’re being asked to find the first partial derivative with respect to π‘₯ 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 π‘₯, we’re interested in saying how the function changes as we let just one of the variables change, as we let π‘₯ 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 π‘₯ with respect to π‘₯ 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 π‘₯ must be zero. And that means the first partial derivative with respect to π‘₯ of our function 𝑓 of π‘₯, 𝑦 is one.

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