1. Demostrar las siguientes derivaciones \( y=3 x^{4}-2 x^{3}+6 x \quad y^{\prime \prime}=36 x^{2}-12 x \) \( y^{\prime}=\sqrt{1+2 x} \quad \) si \( x=4 \quad y^{\prime \prime}=\frac{1}{27} \)

Calcular la derivada de las a) \( F_{(x)}=\left(x^{5} \cdot \operatorname{sen} x\right) \)

(f) Find the constants \( a, b, c \), so that \[ \vec{F}=(x+2 y+a z) \vec{i}+(b x-3 y-z) \vec{j}+(4 x+c y+2 z) \vec{k} \] is irrotational and hence find function \( \phi \) such that \( \vec{F}=\nabla \phi \). (g) Show that the vector field \( \vec{F}=\left(e^{x} \cos y+y z\right) \vec{i}+\left(x z-e^{x} \sin y\right) \vec{j}+(x y+z) \vec{k} \) is conservative by (3 marks) finding a potential for it.

ón \( f_{x}=\frac{1}{2} x^{2}+x \) en el punto de abscisa \( x=a \), PASA \( a=4 \)

Question 3 finding a potential for it. (a) Find the volume of the parallelopiped whose coterminus edges are given by the vectors \( \left.\vec{A}=2 \vec{i}-e^{x} \cos y+y z\right) \vec{i}+\left(x z-e^{x} \sin y\right) \vec{j}+(x y+z) \vec{k} \) is conservative by \( 3 \vec{j}+4 \vec{k}, \vec{B}=\vec{i}+2 \vec{j}-\vec{k} \) and \( \vec{C}=3 \vec{i}-\vec{j}+2 \vec{k} \).

Use Newton's method to find an approximate solution to \( x^{2}-2=0 \), starting from an initia estimate \( x_{1}=2 \). Iterate until 6 iterations. Hints: \( \boldsymbol{f}^{\prime}(\boldsymbol{x})=2 \boldsymbol{x} \) Marks)