Question

Find the differential equation of all tangent lines of the parabola y=x^2

Answer 1

My guess: y=x^2
The derivative of this function is the slope of its tangent line, so:
y'=2x --> slope of the tangent line

y=mx+b
Since y =x^2, and m=y'
x^2=y'x+b
Differentiating wrt x will give:
2x=y'+xy'' ---> answer

Is that right?