Differentiation

**Punch** · August 28th 2011, 06:20 AM

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

**alexmahone** · August 28th 2011, 06:21 AM

Originally Posted by Punch

$\frac{dy}{dx}=2(1+y^2)$

Differentiate both sides twice.

**Punch** · August 28th 2011, 06:26 AM

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

**alexmahone** · August 28th 2011, 06:32 AM

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

$\frac{d^2y}{dx^2}=4y\frac{dy}{dx}$

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

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