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 $-bxy$?
$$\frac{dx}{dt}=ax-bxy$$
You know implicit differentiation, so let's just differentiate our above equation wrt $t$ (assuming $a,b$ constants):
$$\frac{d^2x}{dt^2}=a\frac{dx}{dt}-b\frac{d(xy)}{dt}$$
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 $t$.
That's all the logic there is to your question. Hope it helps!