 # 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:
$\frac{dx}{dt}=ax-bxy$
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 $-bxy$?
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$\frac{dx}{dt}=ax-bxy$
You know implicit differentiation, so let's just differentiate our above equation wrt $t$ (assuming $a,b$ constants):
$\frac{{d}^{2}x}{d{t}^{2}}=a\frac{dx}{dt}-b\frac{d\left(xy\right)}{dt}$
Now use the product rule to find the differentiation of $xy$ wrt $t$. You'll get $dx/dt$ again, but remember to substitute its values from the original expression we have. You'll also get $dy/dt$, but you cannot further simplify as we don't know exactly how $y$ is a function of $t$.

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