# Implicit differentiation; dx/dt

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 $-bxy$?

• y is a function of t? Jan 17 '18 at 1:08
• x and y are both functions of t, start there and remember you know $\frac{dx}{dt}$ Jan 17 '18 at 1:50
• @Srini, yes y is a function of t. Jan 17 '18 at 2:18
• @JacobClaassen Okay, I will try figuring it out! Jan 17 '18 at 2:19

$$\frac{dx}{dt}=ax-bxy$$
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 $t$.