Chain Rule example from lecture

Question

If $$v = \frac{y}{x},$$

then

$$\frac{dy}{dx} = v + x\frac{dv}{dx},$$

and so

$$\frac{d ^ 2y}{dx ^ 2} = 2\frac{dv}{dx} + x\frac{d^2v}{dx^2}.$$

What are the steps to this? I don't see how my professor did it.

Answer 1

Note that $y=x\cdot v$ where $v$ is a function of $x$. Hence, by differentiating the product we find $$\frac{dy}{dx}=\frac{d(x\cdot v)}{dx}=\frac{dx}{dx}\cdot v+x\frac{dv}{dx}=v+x\frac{dv}{dx}.$$ Now we differentiate again and we obtain the next line $$\frac{d^2y}{dx^2}=\frac{d(dy/dx)}{dx}=\frac{dv}{dx}+\left(\frac{dv}{dx}+x\cdot\frac{d^2v}{dx^2}\right)=2\cdot\frac{dv}{dx}+x\cdot\frac{d^2v}{dx^2}.$$