Second Derivative of polar function

Question

i'm trying to find $ \frac{d^2\rho}{(d\phi)^2}$ of $\rho=\tan(\phi+\rho)$ in this exercise is not specified if $\rho$ (argument of tangent) is a constant o another variable.

I do $$\rho'=\sec^2(\phi+\rho) $$ then i re-derivate it $$ \rho''=2 \sec^2(\phi+\rho)\tan(\phi+\rho) $$

In my answer book this must be: $$ −\frac{\cot^3(\phi+\rho)}{\sin(\phi+\rho)}$$

Obviously i make a mistake or a misinterpreted some concept. I need some orientation or a example in order to continue. Thanks in advance.

Answer 1

Well considering $ 0=\tan(\phi+\rho)-\rho $ as a implicit function I proceed to derivate respect to $\phi$ $$ 0=\tan(\phi+\rho)-\rho$$ $$ 0=\sec^2(\phi+\rho)((1+\rho')-\rho'$$ $$ 0=\sec^2(\phi+\rho)+sec^2(\phi+\rho)\rho'-\rho'$$ $$ -\sec^2(\phi+\rho)=\rho'(sec^2(\phi+\rho)-1)$$ $$ \rho'=\frac{-\sec^2(\phi+\rho)}{(sec^2(\phi+\rho)-1)}$$ then I go to the second derivative of $\phi$

$$\rho''=\frac{-(2\sec^2(\phi+\rho)\tan((\phi+\rho))(1+\rho')(\sec^2(\phi+\rho)-1)+(2\sec^2(\phi+\rho)\sec^2(\phi+\rho)\tan(\phi+\rho)(1+\rho')}{(\sec^2(\phi+\rho)-1)^2}$$ $$\rho''=\frac{(1+\rho')[-2\sec^2(\phi+\rho)\tan((\phi+\rho))(\sec^2(\phi+\rho)-1)+(2\sec^4(\phi+\rho)\tan(\phi+\rho]}{(\sec^2(\phi+\rho)-1)^2} $$

$$ \rho''=\frac{(1+\rho')\biggl(-2\sec^2(\phi+\rho)\tan((\phi+\rho))\biggl[(\sec^2(\phi+\rho)-1)+(\sec^2(\phi+\rho)\biggr]\biggr)}{(\sec^2(\phi+\rho)-1)^2} $$

$$ \rho''=\frac{(1+\rho')\biggl(-4\sec^2(\phi+\rho)\tan((\phi+\rho))\biggr)}{(\sec^2(\phi+\rho)-1)} $$ now replacing $\rho'$ in $\rho''$ $$ \rho''=\frac{(1+\frac{-\sec^2(\phi+\rho)}{(sec^2(\phi+\rho)-1)})\biggl(-4\sec^2(\phi+\rho)\tan((\phi+\rho))\biggr)}{(\sec^2(\phi+\rho)-1)} $$

$$ \rho''=\frac{(\frac{-1}{(sec^2(\phi+\rho)-1)})\biggl(-4\sec^2(\phi+\rho)\tan((\phi+\rho))\biggr)}{(\sec^2(\phi+\rho)-1)} $$ Finally $$ \rho''=\frac{\biggl(4\sec^2(\phi+\rho)\tan((\phi+\rho))\biggr)}{(\sec^2(\phi+\rho)-1)^2} $$

this is more tricky than considering the $\rho$ inside of the argument of tangent like a constant, but still not looking like my suppousted answer. Sorry for the long equation but i want to show every step, so if someone find some error or if i miss something somewere please comment.