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

# 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 π(π₯, π¦) = π₯β΄ + 5π₯π¦Β³.

01:45

### Video Transcript

Find the first partial derivative with respect to π₯ of the function π of π₯, π¦ equals π₯ to the fourth power plus five π₯π¦ cubed.

Weβve been given a multivariable function of two variables. Thatβs π₯ and π¦. Weβre being asked to find the first partial derivative with respect to π₯ of this function. Thatβ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 π¦ as a constant and then differentiate as normal.

Letβs do this term by term. Weβ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₯π¦ cubed. Remember, weβre treating π¦ as a constant. So we can treat π¦ cubed and therefore five π¦ cubed as a constant.

And so the derivative of five π₯π¦ cubed here with respect to π₯ is simply five π¦ cubed. And thatβs because we treat π₯ as a power term. Itβ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. Itβs four π₯ cubed plus five π¦ cubed.

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