Question Video: Finding the First Partial Derivative in a Multivariable Function of Two Variables

Video Transcript

Find the first partial derivative with respect to 𝑥 of the function 𝑓 of 𝑥, 𝑦 equals 𝑥 to the fourth power plus five 𝑥𝑦 cubed.

We’ve been given a multivariable function of two variables. That’s 𝑥 and 𝑦. We’re being asked to find the first partial derivative with respect to 𝑥 of this function. That’s often denoted as shown. But what does it mean to find a partial derivative of a function? Well, we want to see how our multivariable function changes if we just let one of those variables change and hold all the others constant. And so to find a partial derivative with respect to 𝑥 of our function, we treat 𝑦 as a constant and then differentiate as normal.

Let’s do this term by term. We’ll begin by differentiating 𝑥 to the fourth power with respect to 𝑥. Well, we know that to differentiate a power term, we multiply the entire term by the exponent, then reduce that exponent by one. So the derivative of 𝑥 to the fourth power with respect to 𝑥 is four 𝑥 cubed. Next, we move on to the second term, five 𝑥𝑦 cubed. Remember, we’re treating 𝑦 as a constant. So we can treat 𝑦 cubed and therefore five 𝑦 cubed as a constant.

And so the derivative of five 𝑥𝑦 cubed here with respect to 𝑥 is simply five 𝑦 cubed. And that’s because we treat 𝑥 as a power term. It’s 𝑥 to the power of one. We multiply the entire term by the exponent and reduce that exponent by one. So we get one times 𝑥 to the power of zero. But, of course, 𝑥 to the power of zero is one. So the derivative of 𝑥 to the power of one is one. And so we found the first partial derivative with respect to 𝑥 of our function. It’s four 𝑥 cubed plus five 𝑦 cubed.