Video: Finding the First Partial Derivative in a Three-Variable Logarithmic Function

Find the first partial derivative with respect to 𝑦 of the function 𝑓(π‘₯, 𝑦, 𝑧) = ln (π‘₯ + 2𝑦 + 3𝑧).

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Video Transcript

Find the first partial derivative with respect to 𝑦 of the function 𝑓 of π‘₯, 𝑦, 𝑧 is equal to the natural log of π‘₯ plus two 𝑦 plus three 𝑧.

Here, we’ve been given a multivariable function. It’s a function defined by three variables. Those are π‘₯, 𝑦, and 𝑧. We’re looking to find its first partial derivative with respect to 𝑦. We represent this as πœ•π‘“ by πœ•π‘¦. When we find the first partial derivative with respect to 𝑦, we treat all the other variables as if they were constants and then differentiate with respect to 𝑦. So, in the expression the natural log of π‘₯ plus two 𝑦 plus three 𝑧, we’re going to treat the variables π‘₯ and 𝑧 as if they are constants.

And so, let’s recall how we differentiate an expression of the form the natural log of π‘Žπ‘₯ plus 𝑏 with respect to π‘₯, where π‘Ž and 𝑏 are constants. The derivative is π‘Ž over π‘Žπ‘₯ plus 𝑏. Note, changing the variable does not change the derivative. We see that the derivative of the natural log of π‘Žπ‘¦ plus 𝑏 with respect to 𝑦 is π‘Ž over π‘Žπ‘¦ plus 𝑏. And so, we’re ready to find the first partial derivative with respect to 𝑦 of our function.

The coefficient of 𝑦 becomes the numerator of our expression. So, the coefficient of 𝑦 is two, and our numerator is two. Then, the denominator is the inner part of our composite function. So, it’s π‘₯ plus two 𝑦 plus three 𝑧. And this means the first partial derivative with respect to 𝑦, that’s πœ•π‘“ πœ•π‘¦, is two over π‘₯ plus two 𝑦 plus three 𝑧.

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