Question

Questão 1:
A derivada de y = ln((x + 2)/(x ^ 2 + 1)) é?
Questão 2:
A derivada da função f(x)=tg (3x²+2x) é?
Questão 3:
Derive y = (log10 x) ^ 3 é?

Answer 1

$\large\boxed{\begin{array}{l}\rm 1)\\\rm y=\ell n\bigg(\dfrac{x+2}{x^2+1}\bigg)\\\\\rm y'=\dfrac{1}{\frac{x+2}{x^2+1}}\cdot\bigg[\dfrac{1\cdot(x^2+1)-(x+2)\cdot2x}{(x^2+1)^2}\bigg]\\\\\rm y'=\dfrac{\diagup\!\!\!\!\!(x^2+\diagup\!\!\!\!1)}{(x+2)}\cdot\bigg[\dfrac{x^2+1-2x^2-4x}{\diagup\!\!\!\!(x^2+\diagup\!\!\!\!1)^2}\bigg]\\\\\rm y'=\dfrac{-x^2-4x+1}{(x+2)\cdot(x^2+1)}\\\huge\boxed{\boxed{\boxed{\boxed{\rm\red{\maltese}~\blue{alternativa~a}}}}}\end{array}}$

$\large\boxed{\begin{array}{l}\rm 2)\\\rm f(x)=tg(3x^4+2x)\\\rm f'(x)=sec^2(3x^4+2x)\cdot(12x^3+2)\\\rm f'(x)=(12x^3+2)\cdot sec^2(3x^4+2x)\\\huge\boxed{\boxed{\boxed{\boxed{\rm\red{\maltese}~\blue{alternativa~d}}}}}\end{array}}$

$\large\boxed{\begin{array}{l}\rm 3)\\\rm y=(\ell og_{10}x)^3\\\rm y'=3\cdot(\ell og_{10}x)^2\cdot\bigg[\dfrac{1}{x\ell n10}\bigg]\\\\\rm y'=\dfrac{3(\ell og_{10}x)^2}{x\cdot\ell n 10}\\\\\huge\boxed{\boxed{\boxed{\boxed{\rm\red{\maltese}~\blue{alternativa~b}}}}}\end{array}}$