### Video Transcript

If three 𝑥 squared plus five is equal to two 𝑥𝑦, prove that 𝑥 d two 𝑦 d𝑥 squared plus two d𝑦 d𝑥 is equal to three.

In order to prove this, we will need to find the second derivative or differentiate the function twice. We will also need to use the product rule, which states that if 𝑦 is equal to 𝑢𝑣, then d𝑦 d𝑥 is equal to 𝑢𝑣 dash plus 𝑣𝑢 dash, where 𝑢 dash and 𝑣 dash are the differential of 𝑢 and 𝑣, respectively.

Our first step is to differentiate each of the three terms individually. Differentiating three 𝑥 squared gives us six 𝑥. Differentiating any constant, in this case five, gives us zero. We need to differentiate the two 𝑥𝑦 term using the product rule. We can let 𝑢 equal two 𝑥 and 𝑣 equal 𝑦. The differential of two 𝑥 is two. Therefore, 𝑢 dash is equal to two. The differential of 𝑦 with respect to 𝑥 is d𝑦 d𝑥. Therefore, 𝑣 dash is equal to d𝑦 d𝑥. Multiplying the 𝑢 and 𝑣 dash terms gives us two 𝑥 d𝑦 d𝑥. Multiplying the 𝑣 and 𝑢 dash terms gives us two 𝑦.

We could try and rearrange to make d𝑦 by d𝑥 the subject at this stage. However, it will be easier to differentiate each of the three terms once again. Differentiating six 𝑥 gives us six. Differentiating the third term, two 𝑦, gives us two d𝑦 d𝑥. In order to differentiate the second term, two 𝑥 d𝑦 d𝑥, we once again need to use the product rule. We can let 𝑢 equal two 𝑥 and 𝑣 equal d𝑦 by d𝑥. Differentiating two 𝑥 gives us two. Therefore, 𝑢 dash is equal to two. Differentiating d𝑦 by d𝑥 in terms of 𝑥 gives us d two 𝑦 d𝑥 squared.

Once again, we need to multiply 𝑢 and 𝑣 dash and then 𝑣 and 𝑢 dash. Multiplying 𝑢 and 𝑣 dash gives us two 𝑥 d two 𝑦 d𝑥 squared. Multiplying 𝑣 and 𝑢 dash gives us two d𝑦 d𝑥. This leaves us with six is equal to two 𝑥 d two 𝑦 d𝑥 squared plus two d𝑦 d𝑥 plus two d𝑦 d𝑥. Grouping the last two terms gives us four d𝑦 d𝑥. We can then divide both sides of the equation by two. Six divided by two is equal to three. Dividing two 𝑥 by two gives us 𝑥. So the second term is 𝑥 multiplied by d two 𝑦 over d𝑥 squared. Dividing the final term by two gives us two multiplied by d𝑦 d𝑥.

We have therefore proved that if three 𝑥 squared plus five is equal to two 𝑥𝑦, then 𝑥 d two 𝑦 d𝑥 squared plus two d𝑦 d𝑥 is equal to three.