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Find the first partial derivative with respect to π₯ of the function π of π₯, π¦ equals π¦ squared times π to the power of negative π₯.
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Here weβve been given a multivariable function.
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Thatβs a function defined by two or more variables.
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Here, those variables are π₯ and π¦.
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Sometimes when dealing with these functions, we want to see what happens when we change just one of the variables.
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This is called finding the first partial derivative of the function.
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We use curly dβs or πβs to represent the first partial derivative.
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And so the first partial derivative with respect to π₯ is ππ by ππ₯.
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And so what we do is we imagine that all the other variables, thatβs the variables that are not π₯, are simply constants.
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Weβre going to treat the variable π¦ as if itβs a constant.
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And weβre going to differentiate some constant times π to the power of negative π₯ with respect to π₯.
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And so we recall how we differentiate an expression of the form ππ to the power of ππ₯ with respect to π₯ for real constants π and π.
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We simply multiply everything by the coefficient of π₯, and everything else remains unchanged.
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And we get ππ times π to the power of ππ₯.
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The coefficient of π₯ in this case is negative one.
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So ππ by ππ₯ is π¦ squared times negative one times π to the power of negative π₯.
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Simplifying, and we find our first partial derivative with respect to π₯ is negative π¦ squared times π to the power of negative π₯.