Math Help - Differentiation

1. Differentiation

Given that $y=tan2x$ and $\frac{dy}{dx}=2(1+y^2)$, show that $\frac{d^3y}{dx^2}=4(\frac{dy}{dx})^2 +4y\frac{d^2y}{dx^2}$

2. Re: Differentiation

Originally Posted by Punch
$\frac{dy}{dx}=2(1+y^2)$
Differentiate both sides twice.

3. Re: Differentiation

Originally Posted by alexmahone
Differentiate both sides twice.
$\frac{d^2y}{dx^2}=2(2y)\frac{dy}{dx}$

$\frac{d^3y}{dx^2}=4y\frac{d^2y}{dx^2}+\frac{dy}{dx }(4)$

But the answer has a square for dy/dx

4. Re: Differentiation

Originally Posted by Punch
$\frac{d^2y}{dx^2}=2(2y)\frac{dy}{dx}$

$\frac{d^3y}{dx^2}=4y\frac{d^2y}{dx^2}+\frac{dy}{dx }(4)$

But the answer has a square for dy/dx
$\frac{d^2y}{dx^2}=4y\frac{dy}{dx}$

$\frac{d^3y}{dx^2}=4\left(y\frac{d^2y}{dx^2}+\frac{ dy}{dx}\frac{dy}{dx}\right)=4(\frac{dy}{dx})^2+4y \frac{d^2y}{dx^2}$