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Find the first partial derivative with respect to ๐ฅ of the function ๐ of ๐ฅ, ๐ฆ equals ๐ฅ plus two ๐ฆ.
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This is a multivariable function, a function in terms of more than one variable.
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So here, thatโs ๐ฅ and ๐ฆ.
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Weโre being asked to find the first partial derivative with respect to ๐ฅ of our function.
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So we denote that as shown using these sort of swirly dโs.
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Essentially, when weโre finding the first partial derivative with respect to ๐ฅ, weโre interested in saying how the function changes as we let just one of the variables change, as we let ๐ฅ change.
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So we hold all the other variables constant.
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So letโs see what that might look like.
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And weโre going to treat this term by term.
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The derivative of ๐ฅ with respect to ๐ฅ is just one.
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Now, remember, weโre holding ๐ฆ constant.
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So we differentiate as if ๐ฆ is a constant.
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And then the derivative of two ๐ฆ with respect to ๐ฅ must be zero.
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And that means the first partial derivative with respect to ๐ฅ of our function ๐ of ๐ฅ, ๐ฆ is one.