I have been given the following first order derivative equation: (dx)/(dt)=ax−bxy How do I find the second derivative? I know how to find implicit derivatives but since the differential in this case needs to be found in terms of dx/dt, instead of the dy/dx which I am used to, I am confused. Should I used the product rule to find the derivative of −bxy?

Urraiyg 2022-09-19 Answered
I have been given the following first order derivative equation:
d x d t = a x b x y
How do I find the second derivative? I know how to find implicit derivatives but since the differential in this case needs to be found in terms of d x / d t, instead of the d y / d x which I am used to, I am confused. Should I used the product rule to find the derivative of b x y?
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Answers (1)

AKPerqk
Answered 2022-09-20 Author has 9 answers
d x d t = a x b x y
You know implicit differentiation, so let's just differentiate our above equation wrt t (assuming a , b constants):
d 2 x d t 2 = a d x d t b d ( x y ) d t
Now use the product rule to find the differentiation of x y wrt t. You'll get d x / d t again, but remember to substitute its values from the original expression we have. You'll also get d y / d t, but you cannot further simplify as we don't know exactly how y is a function of t.

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