Find the first partial derivative with respect to 𝑦 of the function 𝑓 of 𝑥, 𝑦 equals 𝑥 of the fourth power.
Now, this 𝑓 of 𝑥, 𝑦 indicates to us we have a multivariable function. It’s a function defined by two variables, 𝑥 and 𝑦. Sometimes when working with multivariable functions, we want to see how the function changes when we just vary one of its variables. This is called finding the first partial derivative. We use these curly d’s or 𝜕 to represent the first partial derivative.
So the first partial derivative with respect to 𝑦 is 𝜕𝑓 𝜕𝑦. Now, what we do here is we differentiate as normal. But we treat all the other variables as constants. So in this case, we’re going to vary 𝑦 and treat 𝑥 as a constant. But of course, when we differentiate a constant, we simply get zero. So treating 𝑥 to the fourth power as a constant and differentiating with respect to 𝑦 gives us 𝜕𝑓 𝜕𝑦 equals zero. And so we’re done. The first partial derivative with respect to 𝑦 of the function 𝑓 of 𝑥, 𝑦 equals 𝑥 to the fourth power is zero.