Regla de la cadena multivariable

Sea la función

 $z=e^{x-3y}$  donde  $x=vu^3-v$  como  $y=u-4v$  las derivadas  $\frac{\partial z}{\partial u}$  y  $\frac{\partial z}{\partial v}$  es:

•  $\frac{\partial z}{\partial u}=3e^{x-3y}\left(vu^2-1\right);\ \frac{\partial z}{\partial v}=e^{x-3y}\left(u^3+11\right)$  

•  $\frac{\partial z}{\partial u}=3e^{x-3y}\left(u^3+11\right);\ \frac{\partial z}{\partial v}=e^{x-3y}\left(vu^2-1\right)$  

•  $\frac{\partial z}{\partial u}=3e^{x-3y}\left(u^3-11\right);\ \frac{\partial z}{\partial v}=e^{x-3y}\left(vu^2+1\right)$  

Sea la función

 $z=y^2\sqrt{x+1}$  donde  $x=t^3-t$  como  $y=t^2-2t+4$  la derivada  $\frac{\partial z}{\partial t}$ es:

•  $\frac{\text{d}z}{\text{d}t}=\frac{y^2}{2\sqrt{x+1}}\left(3t^2-1\right)+2y\sqrt{x+1}\left(2t-2\right)$  

•  $\frac{\text{d}z}{\text{d}t}=\frac{y^2}{2\sqrt{x+1}}\left(3t^2-1\right)+2y\sqrt{x+1}\left(2t+2\right)$  

Sea la función

 $f=x+\left(y-2\right)^3$  donde  $x=r+5t$  como  $y=3r-4t$  la derivada  $\frac{\partial f}{\partial t}$  es:

•  $\frac{\partial f}{\partial t}=5-12\left(y-2\right)^2$  

•  $\frac{\partial f}{\partial t}=5+12\left(y-2\right)^2$  

•  $\frac{\partial f}{\partial t}=5-12\left(y+2\right)^2$  

•  $\frac{\partial f}{\partial t}=5+12\left(y+2\right)^2$  

Sea la función

 $z=\sqrt{x+y^2}$  donde  $x=t^2+t$  como  $y=t-200$  la derivada  $\frac{\partial z}{\partial t}$  es:

•  $\frac{\partial z}{\partial t}=\frac{2t^2+4t+1}{2\sqrt{x+y^2}}$  

•  $\frac{\partial z}{\partial t}=\frac{2t^2-4t+1}{2\sqrt{x+y^2}}$  

•  $\frac{\partial z}{\partial t}=\frac{2t^2+4t+1}{2\sqrt{x-y^2}}$  

•  $\frac{\partial z}{\partial t}=\frac{2t^2+t+1}{2\sqrt{x+y^2}}$  

Sea la función

 $z=e^{xyz}$  donde  $x=r+5t$  como  $y=2t^3$  además  $z=1-3r^2$   las derivadas  $\frac{\partial z}{\partial t}$  y  $\frac{\partial z}{\partial r}$  es:

•  $\frac{\partial z}{\partial t}=5yze^{xyz}+xze^{xyz}6t^2;\ \frac{\partial z}{\partial r}=yze^{xyz}6xye^{xyz}r$  

•  $\frac{\partial z}{\partial t}=5yze^{xyz}-xze^{xyz}6t^2;\ \frac{\partial z}{\partial r}=yze^{xyz}6xye^{xyz}r$  

Sea la función

 $y=e^{2x^2}$  donde  $x=r-3t^2$  las derivadas  $\frac{\partial y}{\partial r}$  y  $\frac{\partial y}{\partial t}$  es:

•  $\frac{\partial y}{\partial r}=4xe^{2x^2};\ \frac{\partial y}{\partial t}=-24txe^{2x^2}$  

•  $\frac{\partial y}{\partial r}=4xe^{2x^2};\ \frac{\partial y}{\partial t}=24txe^{2x^2}$  

•  $\frac{\partial y}{\partial r}=-4xe^{2x^2};\ \frac{\partial y}{\partial t}=-24txe^{2x^2}$  

•  $\frac{\partial y}{\partial r}=-4xe^{2x^2};\ \frac{\partial y}{\partial t}=24txe^{2x^2}$  

Sea la función

 $w=xy-x^2yz$  donde  $x=t-1$  como  $y=2t^3$ además  $z=t^2+1$   la derivada  $\frac{\partial w}{\partial t}$  es:

•  $\frac{\text{d}w}{\text{d}t}=\left(y-2xyz\right)+\left(x-z-x^2\right)6t^2-\left(y+x^2y\right)2t$  

•  $\frac{\text{d}w}{\text{d}t}=\left(y-2xyz\right)-\left(x-z-x^2\right)6t^2-\left(y+x^2y\right)2t$