Find 𝑑𝑦 by 𝑑𝑥 if 𝑦 is equal to 𝑥 squared plus three over 𝑥 cubed plus three.
Here we have to differentiate a rational function and so we need to use the quotient rule for differentiation which allows us to write the derivative of a quotient of two functions 𝑓 of 𝑥 over 𝑔 of 𝑥 in terms of 𝑓 of 𝑥 𝑔 of 𝑥 and their derivatives.
Let’s compare 𝑓 of 𝑥 over 𝑔 of 𝑥 to what we have to differentiate to see what 𝑓 of 𝑥 and 𝑔 of 𝑥 are. 𝑓 of 𝑥 is the numerator 𝑥 squared plus three and 𝑔 of 𝑥 is the denominator 𝑥 cubed plus three. To apply the quotient rule, we also need the derivatives of 𝑓 of 𝑥 and 𝑔 of 𝑥. The derivative of 𝑥 squared plus three is two 𝑥, which we can get by using the fact that the derivative of a power of 𝑥, 𝑥 to the 𝑛, with respect to 𝑥 is 𝑛 times 𝑥 to the 𝑛 minus one and the derivative of a constant function 𝑐 with respect to 𝑥 is zero.
Applying these rules again, we find that the derivative of 𝑔 of 𝑥 with respect to 𝑥 is three 𝑥 squared. Now, we have all the ingredients needed to apply the quotient rule. We substitute 𝑥 squared plus three for 𝑓 of 𝑥 and 𝑥 cubed plus three for 𝑔 of 𝑥 on the left-hand side. Continuing to substitute, the first term in the numerator becomes 𝑥 cubed plus three times two 𝑥. And from this, we subtract 𝑥 squared plus three times three 𝑥 squared and we divide this by 𝑥 cubed plus three squared.
Now let’s expand and simplify in the numerator. Expanding, we get two 𝑥 to the four plus six 𝑥 minus three 𝑥 to the four minus nine 𝑥 squared. And we see that there are some like terms that we can combine to get our final answer minus 𝑥 to the four minus nine 𝑥 squared plus six 𝑥 all over 𝑥 cubed plus three squared.