Question

Calcula las derivadas de las siguientes funciones utilizando las propiedades de la suma, el producto o el cociente de derivadas:
a. f(x) = 3x^2 + sen x
b. f(x) = In x • (x^2+2x+1)
c. f(x) = x •cos x/ raíz cuadrada de x+1
d. f(x) = cos x/ raíz cuadrada de x+1

Answer 1

Hola.

$Regla \ de \ la \ suma: \boxed{\left(f\pm g\right)^'} = \boxed{f' \pm g'}$

$Regla \ del \ producto: \boxed{\left(f\cdot g\right)^'} = \boxed{f^'\cdot g+f\cdot g^'}$

Regla del cociente: $\boxed{\left(\frac{f}{g}\right)^'=\frac{f^'\cdot g-g^'\cdot f}{g^2}}$

$a) \frac{d}{dx}\left(3x^2+\sin \left(x\right)\right) \\ \\ Aplicamos \ regla \ de \ la \ suma: \boxed{\left(f\pm g\right)^'} = \boxed{f' \pm g'} \\ \\ \frac{d}{dx}\left(3x^2\right)+\frac{d}{dx}\left(\sin \left(x\right)\right) \\ \\ 3\frac{d}{dx}\left(x^2\right) \\ \\ 3\cdot \:2x^{2-1} = 6x \\ \\ \frac{d}{dx}\left(\sin \left(x\right)\right) = cosx \\ \\ R1: 6x+\cos \left(x\right)$

$b) \frac{d}{dx}\left(Inx\left(x^2+2x+1\right)\right) \\ \\ In\frac{d}{dx}\left(x\left(x^2+2x+1\right)\right) \\ \\ Aplicamos \ regla \ del \ producto: \\ \\ In\left(\frac{d}{dx}\left(x\right)\left(x^2+2x+1\right)+\frac{d}{dx}\left(x^2+2x+1\right)x$
$In\left(1\cdot \left(x^2+2x+1\right)+\left(2x+2\right)x\right) \\ \\ nI\left(x^2+2x+x\left(2x+2\right)+1\right) \\ \\ x^2+2x+1+2x^2+2x \\ \\ x^2+2x^2+2x+2x+1 \\ \\ 3x^2+4x+1 \\ \\ In\left(3x^2+4x+1\right) \\ \\ R2: In\left(3x^2+4x+1\right)$


$c) \frac{d}{dx}\left(x\cos \left(\frac{x}{\sqrt{x+1}}\right)\right) \\ \\ Aplicamos \ regla \ del \ producto: \frac{d}{dx}\left(x\right)\cos \left(\frac{x}{\sqrt{x+1}}\right)+\frac{d}{dx}\left(\cos \left(\frac{x}{\sqrt{x+1}}\right)\right)x$ 
$\frac{d}{dx}\left(x\right)=1 \\ \\ Aplicamos \ regla \ de \ la \ cadena: \ \frac{df\left(u\right)}{dx}=\frac{df}{du}\cdot \frac{du}{dx} \\ \\ Sustituimos: \ \frac{x}{\sqrt{x+1}}\mathrm{=}u \\ \\ \frac{d}{du}\left(\cos \left(u\right)\right)\frac{d}{dx}\left(\frac{x}{\sqrt{x+1}}\right) \\ \\ \frac{d}{du}\left(\cos \left(u\right)\right) = -sen(u) \\ \\ \frac{d}{dx}\left(\frac{x}{\sqrt{x+1}}\right)$
$Aplicamos \ regla \ del \ cociente: \frac{\frac{d}{dx}\left(x\right)\sqrt{x+1}-\frac{d}{dx}\left(\sqrt{x+1}\right)x}{\left(\sqrt{x+1}\right)^2} \\ \\ = \frac{1}{2\sqrt{u}}\cdot \:1 \\ \\ Sustituimos (u) = \frac{1}{2\sqrt{x+1}}\cdot \:1 \\ \\ \frac{1\cdot \sqrt{x+1}-\frac{1}{2\sqrt{x+1}}x}{\left(\sqrt{x+1}\right)^2} \\ \\ \frac{\sqrt{x+1}-\frac{1}{2\sqrt{x+1}}x}{\left(\sqrt{x+1}\right)^2} \\ \\ \\ \left(\sqrt{x+1}\right)^2 = x+1 \\ \\ \frac{\sqrt{x+1}-\frac{1}{2\sqrt{x+1}}x}{x+1}$
$\frac{1}{2\sqrt{x+1}}x \\ \\ \frac{x}{2\sqrt{x+1}} \\ \\ = \frac{\sqrt{x+1}-\frac{x}{2\sqrt{x+1}}}{x+1} \\ \\ \frac{\sqrt{x+1}}{1}-\frac{x}{2\sqrt{x+1}} \\ \\ MCM: 2\sqrt{x+1} \\ \\ Reescribimos: \frac{\sqrt{x+1}\cdot \:2\sqrt{x+1}}{2\sqrt{x+1}}-\frac{x}{2\sqrt{x+1}} \\ \\ combinamos: \ \frac{2\sqrt{x+1}\sqrt{x+1}-x}{2\sqrt{x+1}} \\ \\ 2\sqrt{x+1}\sqrt{x+1}-x \\ \\ Aplicamos: \left(x+1\right)^{\frac{1}{2}+\frac{1}{2}}=\:\left(x+1\right)^1=\:x+1 \\ \\ =2\left(x+1\right)- x$
$\frac{2\left(x+1\right)-x}{2\sqrt{x+1}} \\ \\ 2\left(x+1\right) \\ \\ 2\cdot \:x+2\cdot \:1 \\ \\ 2x+2-x \\ \\ x+2 \\ \\ \frac{x+2}{2\sqrt{x+1}} \\ \\ \frac{\frac{x+2}{2\sqrt{x+1}}}{x+1} \\ \\ \frac{x+2}{2\left(x+1\right)\sqrt{x+1}} \\ \\ Reescribimos: \ \left(-\sin \left(u\right)\right)\frac{x+2}{2\left(x+1\right)\sqrt{x+1}} \\ \\ Sustituimos: u=\frac{x}{\sqrt{x+1}} \\ \\ \left(-\sin \left(\frac{x}{\sqrt{x+1}}\right)\right)\frac{x+2}{2\left(x+1\right)\sqrt{x+1}}$
$-\sin \left(\frac{x}{\sqrt{x+1}}\right)\frac{x+2}{2\left(x+1\right)\sqrt{x+1}} \\ \\ Multiplicamos: \ -\frac{\left(x+2\right)\sin \left(\frac{x}{\sqrt{x+1}}\right)}{2\left(x+1\right)\sqrt{x+1}} \\ \\ 1\cdot \cos \left(\frac{x}{\sqrt{x+1}}\right)+\left(-\frac{\sin \left(\frac{x}{\sqrt{x+1}}\right)\left(x+2\right)}{2\left(x+1\right)\sqrt{x+1}}\right)x$
$\cos \left(\frac{x}{\sqrt{x+1}}\right)-\frac{\sin \left(\frac{x}{\sqrt{x+1}}\right)\left(x+2\right)}{2\left(x+1\right)\sqrt{x+1}}x \\ \\ \frac{\sin \left(\frac{x}{\sqrt{x+1}}\right)\left(x+2\right)}{2\left(x+1\right)\sqrt{x+1}}x$

$R3: \cos \left(\frac{x}{\sqrt{x+1}}\right)-\frac{x\sin \left(\frac{x}{\sqrt{x+1}}\right)\left(x+2\right)}{2\left(x+1\right)\sqrt{x+1}}$

d) Es demasiado larga....

¡Espero haberte ayudado, saludos...!