Given that four 𝑥 cubed minus two 𝑦 cubed plus 18 equals zero, find d𝑦 d𝑥.
So what we’re gonna do here is actually differentiate our function. And to do that, what we’re gonna use is implicit differentiation. Now the first stage of differentiating implicitly is actually to differentiate each term in our function with respect to 𝑥. Now if we’re gonna differentiate each term with respect to 𝑥, this is gonna be very straightforward for the terms that just include 𝑥 or just a numerical value. However, I’m just gonna show you it’s slightly different when we’re dealing with the one which is in 𝑦.
Well, our first term is just gonna be 12𝑥 squared. Then I’m gonna leave our second term just as minus 𝑑 d𝑥 of two 𝑦 cubed cause what I wanna do is deal with this separately. And then, we get plus zero cause if you differentiate 18, you just get zero and then equals zero because again if you differentiate zero, you just get zero.
Okay great, so now let’s differentiate our 𝑦 term with respect to 𝑥. So we can now explain how we’re gonna differentiate our term two 𝑦 cubed with respect to 𝑥. So first of all, if we have a function that’s in 𝑦 or a term that’s in 𝑦 and we want to differentiate it with respect to 𝑥, we can actually apply the chain rule and say that it’s gonna be equal to the same function differentiated with respect to 𝑦 multiplied by d𝑦 d𝑥.
So now what we’re gonna do is actually apply it to our term in 𝑦. So we we’re gonna have the derivative of two 𝑦 cubed with respect to 𝑥. It’s gonna be equal to six 𝑦 squared. And that’s because if we actually differentiate two 𝑦 cubed with respect to 𝑦, we get six 𝑦 squared because we multiplied the exponent three by the coefficient two gives us six. And then we reduce the exponent by one gives us 𝑦 squared. And then, this is multiplied by d𝑦 d𝑥. So great, we now know that derivative with respect to 𝑥 of two 𝑦 cubed is equal to six 𝑦 squared d𝑦 d𝑥.
So now what we’re gonna do is move on to the second stage of our implicit differentiation. And what we do now is we’re actually gonna rearrange to make d𝑦 d𝑥 the subject because this is what we’re looking for.
So the first thing we’re gonna do is actually subtract 12𝑥 squared from each side. So we get negative six 𝑦 squared d𝑦 d𝑥 is equal to negative 12𝑥 squared. And then next, what I do is actually divide through by negative six 𝑦 squared to leave d𝑦 d𝑥 on its own. So we get d𝑦 d𝑥 is equal to negative 12𝑥 squared over negative six 𝑦 squared. So then, we actually divide the numerator and denominator by negative six. So we’re left with two 𝑥 squared over 𝑦 squared.
So therefore, we can say that given that four 𝑥 cubed minus two 𝑦 cubed plus 18 equals zero, then, d𝑦 d𝑥 is gonna be equal to two 𝑥 squared over 𝑦 squared.