Find the first partial derivative with respect to 𝑦 of 𝑓 of 𝑥, 𝑦 equals 𝑥 squared 𝑦 minus three 𝑦 to the fourth power.
We’ve been given a multivariable function. That’s a function in terms of both 𝑥 and 𝑦. And we’re being asked to find the first partial derivative with respect to 𝑦 of the function. We use curly d’s as shown to denote this. Now, when a function is made up of multiple variables, we look to see how the function changes as we let just one of those variables change. And we hold all the others constant.
In this case, we’re going to let 𝑦 change. And so we treat the other variable 𝑥 as a constant. Aside from this, the rules for differentiation are roughly the same. Let’s go term by term. Let’s differentiate 𝑥 squared 𝑦.
Since we’re imagining 𝑥 as a constant, we can also imagine 𝑥 squared to be a constant. And when we differentiate a constant times 𝑦 with respect to 𝑦, we’re just left with that constant. And so the first part of our partial derivative is just 𝑥 squared. Next, we differentiate negative three 𝑦 to the fourth power. This is simply a power term. So we differentiate as normal. We multiply the entire term by the exponent and then reduce that exponent by one. So it’s negative four times three 𝑦 cubed. This partial derivative simplifies to 𝑥 squared minus 12𝑦 cubed. And so we found the first partial derivative with respect to 𝑦 of our function.
And of course, we could have done this with respect to 𝑥. This time, though, we would be treating 𝑦 as a constant. So the derivative of 𝑥 squared with respect to 𝑥 is two 𝑥. And that means if we treat 𝑦 as a constant, the first part of our first partial derivative with respect to 𝑥 of our function is two 𝑥𝑦. Then, since we would be treating 𝑦 as a constant, 𝑦 to the fourth power, and therefore negative three 𝑦 to the fourth power, is also a constant. And we know when we differentiate a constant with respect 𝑥, we get zero. And so the first partial derivative with respect to 𝑥 of our function would have been two 𝑥𝑦 had that been required.