Question Video: Differentiating Rational Functions Using the Quotient Rule Mathematics • Higher Education

Video Transcript

Find the first derivative of 𝑦 is equal to 𝑥 minus 93 all divided by 𝑥 plus 13.

We need to find the first derivative of 𝑦 is equal to the quotient of two functions. We can see 𝑦 is a function of 𝑥. This means we need to find the derivative of 𝑦 with respect to 𝑥. And there are, in fact, a few different ways we could approach this. For example, we could use algebraic division to rewrite our expression for 𝑦. We could then evaluate the derivative of this expression by using the quotient rule, the chain rule, or the general power rule, and this would work. However, in this case, we’re just going to evaluate the derivative of this expression by using the quotient rule.

So to evaluate this derivative, we need to recall the quotient rule. The quotient rule tells us the derivative of the quotient of two functions 𝑢 over 𝑣 is equal to 𝑢 prime times 𝑣 minus 𝑣 prime times 𝑢 all divided by 𝑣 squared. We’ll set 𝑢 of 𝑥 to be the function in our numerator, that’s 𝑥 minus 93, and 𝑣 of 𝑥 to be the function in our denominator, that’s 𝑥 plus 13. We see to apply the quotient rule, we need to find expressions for 𝑢 prime and 𝑣 prime.

Let’s start with 𝑢 prime of 𝑥. That’s the derivative of the linear function 𝑥 minus 93 with respect to 𝑥. Of course, this is a linear function, so its slope will be equal to the coefficient of 𝑥, which in this case is one. And of course, we can do exactly the same thing to find 𝑣 prime of 𝑥. It’s the derivative of the linear function 𝑥 plus 13 with respect to 𝑥, which will be the coefficient of 𝑥, which is also one. We’re now ready to find an expression for d𝑦 by d𝑥 by using the quotient rule. It’s equal to 𝑢 prime of 𝑥 times 𝑣 of 𝑥 minus 𝑣 prime of 𝑥 times 𝑢 of 𝑥 all divided by 𝑣 of 𝑥 all squared.

Substituting in our expressions for 𝑢 of 𝑥, 𝑣 of 𝑥, 𝑢 prime of 𝑥, and 𝑣 prime of 𝑥, we get that d𝑦 by d𝑥 is equal to one times 𝑥 plus 13 minus one times 𝑥 minus 93 all divided by 𝑥 plus 13 all squared. And, of course, we can start simplifying. First, multiplying by one doesn’t change any of the values in our numerator. And this leaves us with a new numerator of 𝑥 plus 13 minus 𝑥 minus 93. Next, we’ll distribute the negative one over our parentheses. And if we do this, we get negative 𝑥 plus 93.

So now the numerator of our expression is 𝑥 plus 13 minus 𝑥 plus 93, and we can simplify even more. We can cancel 𝑥 minus 𝑥 to give us zero. And finally, in our numerator, 13 plus 93 is equal to 106. So we’ve shown d𝑦 by d𝑥 is equal to 106 divided by 𝑥 plus 13 all squared, and this is our final answer. Therefore, by using the quotient rule, we were able to show if 𝑦 is equal to 𝑥 minus 93 all divided by 𝑥 plus 13, then d𝑦 by d𝑥 is equal to 106 divided by 𝑥 plus 13 all squared.