Build a shifted-tangent expansion and measure its error

Given $y=\tan(\pi/4+x)$, show that $\dfrac{d^2y}{dx^2}=2y\dfrac{dy}{dx}$. Find its series through $x^4$. When $x=0.1$, $y=1.22304888$ correct to 8 decimal places. Verify that the fourth-degree estimate has error about $0.004\%$.