Free Printable Regla de la cadena multivariable

1.

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:

a)

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

b)

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

c)

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

2.

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:

a)

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

b)

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

3.

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:

a)

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

b)

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

c)

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

d)

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

4.

Sea la función

$z=x^2+xy-y^3$ donde $x=r^2-5s$ como $y=s^3$ las derivadas $\frac{\partial z}{\partial r}$ y $\frac{\partial z}{\partial s}$ es:

a)

$\frac{\partial z}{\partial r}=\left(2x+y\right)2r;\ \frac{\partial z}{\partial s}=-5\left(2x+y\right)+\left(y-3y^2\right)3s^2$

b)

$\frac{\partial z}{\partial r}=\left(2x+y\right)2r;\ \frac{\partial z}{\partial s}=5\left(2x+y\right)+\left(y-3y^2\right)3s^2$

5.

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:

a)

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

b)

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

c)

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

d)

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

6.

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:

a)

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

b)

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

7.

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:

a)

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

b)

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

c)

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

d)

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

8.

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:

a)

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

b)

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

Regla de la cadena multivariable

Total questions: 8

Worksheet time: 2hrs 0mins

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