# Video: Differentiating Rational Functions Using the Quotient Rule

Find the first derivative of the function 𝑦 = (4𝑥² + 5𝑥 + 5)/(4𝑥² − 2𝑥 + 3).

01:47

### Video Transcript

Find the first derivative of the function 𝑦 is equal to four 𝑥 squared plus five 𝑥 plus five all over four 𝑥 squared minus two 𝑥 plus three.

We can see that our function is a rational function. Therefore, we can use the quotient rule in order to find the derivative. The quotient rule tells us that 𝑢 over 𝑣 dash is equal to 𝑣𝑢 dash minus 𝑢𝑣 dash all over 𝑣 squared. Where 𝑢 is the numerator of our function and 𝑣 is the denominator. In our case, 𝑢 is equal to four 𝑥 squared plus five 𝑥 plus five. And 𝑣 is equal to four 𝑥 squared minus two 𝑥 plus three. Now, we must find 𝑢 prime and 𝑣 prime. We do this by differentiating 𝑢 and 𝑣 with respect to 𝑥.

Since 𝑢 and 𝑣 are both polynomial functions, we can find their derivatives by taking each term and multiplying the term by the power of 𝑥. And then, decreasing the power of 𝑥 by one. And doing this, we find that 𝑢 prime is equal to eight 𝑥 plus five. And 𝑣 prime is equal to eight 𝑥 minus two. Substituting these into our formula, we find that the first derivative of 𝑦 or 𝑦 prime is equal to 𝑣 multiplied by 𝑢 prime minus 𝑢 multiplied by 𝑣 prime all over 𝑣 squared. Now, the result here looks quite daunting. However, we can still expand the brackets and then simplify. This is what we obtain after expanding the brackets in the numerator.

Our final step is to simplify the numerator. Now, we have reached our solution. Which is that the first derivative of 𝑦 or 𝑦 prime is equal to negative 28𝑥 squared minus 16𝑥 plus 25 all over four 𝑥 squared minus two 𝑥 plus three all squared.