Question

derivada de f(x)=√x+1 quero a resolução.

Answer 1

$f(x)=\sqrt{x+1}\\\\ f(x)=(x+1)^{1/2}$


Esta é uma função composta. Podemos enxergar dessa forma:

$\left\{ \begin{array}{l} f(x)=g(x)^{1/2}\\\\ g(x)=x+1 \end{array} \right.$


Então, vamos derivar usando a Regra da Cadeia:

$f'(x)=\left[g(x)^{1/2} \right ]'\\\\ f'(x)=\dfrac{1}{2}\,g(x)^{(1/2)-1}\cdot g'(x)\\\\\\ f'(x)=\dfrac{1}{2}\,g(x)^{-1/2}\cdot g'(x)\\\\\\ f'(x)=\dfrac{1}{2}\,(x+1)^{-1/2}\cdot (x+1)'\\\\\\ f'(x)=\dfrac{1}{2}\cdot \dfrac{1}{(x+1)^{1/2}}\cdot (1+0)\\\\\\ f'(x)=\dfrac{1}{2}\cdot \dfrac{1}{\sqrt{x+1}}\cdot 1\\\\\\ \therefore~~\boxed{\begin{array}{c} f'(x)=\dfrac{1}{2\sqrt{x+1}} \end{array}}$

Answer 2

f(x)=√x+1
f(x) = (x + 1)¹/²
f'(x) = 1/2(x + 1)¹/²⁻¹. 1
f'(x) = 1/2( x + 1)⁻¹/²
                      1
f'(x) =    _____________
                2√ x + 1