Find the first partial derivative of the function 𝑓 of 𝑥, 𝑦, 𝑧 equals 𝑥 to the power of 𝑦 over 𝑧 with respect to 𝑥.
In this question, we’ve been given a multivariable function. It’s a function defined by three variables: 𝑥, 𝑦, and 𝑧. Now, sometimes when our function is made up of multiple variables, we’re interested in how the function changes as we let just one of the variables change and hold all the others constant. This question is asking us to find the first partial derivative of our function with respect to 𝑥. So, we’re going to be essentially letting 𝑥 change and keeping 𝑦 and 𝑧 constant. We use curly d’s to define the first partial derivative. And when we’re finding the first partial derivative of 𝑓 with respect to 𝑥, it looks like this. And we pronounce that 𝜕𝑓 𝜕 𝑥.
Now, we’re going to be holding 𝑦 and 𝑧 constant. So, 𝑦 over 𝑧 itself, we can imagine to be a constant. And so, we recall that to find the derivative of a power term like 𝑎𝑥 to the 𝑛th power, we multiply the entire term by that power or exponent and reduce that exponent by one. In this case, then, we multiply our entire term by 𝑦 over 𝑧 and then reduce 𝑦 over 𝑧 by one. And so, the first partial derivative of the function with respect to 𝑥 is 𝑦 over 𝑧 times 𝑥 to the power of 𝑦 over 𝑧 minus one.