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

Find the first partial derivative with respect to π‘₯ of the function 𝑓(π‘₯, 𝑦) = π‘₯⁴ + 5π‘₯𝑦³.

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Video Transcript

Find the first partial derivative with respect to π‘₯ of the function 𝑓 of π‘₯, 𝑦 equals π‘₯ to the fourth power plus five π‘₯𝑦 cubed.

We’ve been given a multivariable function of two variables. That’s π‘₯ and 𝑦. We’re being asked to find the first partial derivative with respect to π‘₯ 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 π‘₯ 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 π‘₯ to the fourth power with respect to π‘₯. 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 π‘₯ to the fourth power with respect to π‘₯ is four π‘₯ cubed. Next, we move on to the second term, five π‘₯𝑦 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 π‘₯𝑦 cubed here with respect to π‘₯ is simply five 𝑦 cubed. And that’s because we treat π‘₯ as a power term. It’s π‘₯ to the power of one. We multiply the entire term by the exponent and reduce that exponent by one. So we get one times π‘₯ to the power of zero. But, of course, π‘₯ to the power of zero is one. So the derivative of π‘₯ to the power of one is one. And so we found the first partial derivative with respect to π‘₯ of our function. It’s four π‘₯ cubed plus five 𝑦 cubed.

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