5.	En cada caso encontrar las derivadas parciales ∂w , ∂w
 






y ∂w .
 
∂x	∂y	∂z
 
a)	w = exyz + arctan( 3xy )
b)	w = x+z
c)	w = xyz + √xy + z
 
d)	w = cos zexyz
e)	w = x2y5z7 − x+y3−z5
f ) w = ln (xy + z2)


Vamos a encontrar las derivadas parciales para cada una de las funciones \( w(x, y, z) \) dadas en el problema:

### a) \( w = e^{xyz} + \arctan(3xy) \)

1. **\(\frac{\partial w}{\partial x}\)**:
   \[
   \frac{\partial}{\partial x}(e^{xyz}) = yze^{xyz}
   \]
   \[
   \frac{\partial}{\partial x}(\arctan(3xy)) = \frac{3y}{1 + (3xy)^2}
   \]
   Por lo tanto:
   \[
   \frac{\partial w}{\partial x} = yze^{xyz} + \frac{3y}{1 + (3xy)^2}
   \]

2. **\(\frac{\partial w}{\partial y}\)**:
   \[
   \frac{\partial}{\partial y}(e^{xyz}) = xze^{xyz}
   \]
   \[
   \frac{\partial}{\partial y}(\arctan(3xy)) = \frac{3x}{1 + (3xy)^2}
   \]
   Por lo tanto:
   \[
   \frac{\partial w}{\partial y} = xze^{xyz} + \frac{3x}{1 + (3xy)^2}
   \]

3. **\(\frac{\partial w}{\partial z}\)**:
   \[
   \frac{\partial}{\partial z}(e^{xyz}) = xye^{xyz}
   \]
   \[
   \frac{\partial}{\partial z}(\arctan(3xy)) = 0
   \]
   Por lo tanto:
   \[
   \frac{\partial w}{\partial z} = xye^{xyz}
   \]

---

### b) \( w = x + z \)

1. **\(\frac{\partial w}{\partial x}\)**:
   \[
   \frac{\partial w}{\partial x} = 1
   \]

2. **\(\frac{\partial w}{\partial y}\)**:
   \[
   \frac{\partial w}{\partial y} = 0
   \]

3. **\(\frac{\partial w}{\partial z}\)**:
   \[
   \frac{\partial w}{\partial z} = 1
   \]

---

### c) \( w = xyz + \sqrt{xy} + z \)

1. **\(\frac{\partial w}{\partial x}\)**:
   \[
   \frac{\partial}{\partial x}(xyz) = yz
   \]
   \[
   \frac{\partial}{\partial x}(\sqrt{xy}) = \frac{y}{2\sqrt{xy}}
   \]
   \[
   \frac{\partial}{\partial x}(z) = 0
   \]
   Por lo tanto:
   \[
   \frac{\partial w}{\partial x} = yz + \frac{y}{2\sqrt{xy}}
   \]

2. **\(\frac{\partial w}{\partial y}\)**:
   \[
   \frac{\partial}{\partial y}(xyz) = xz
   \]
   \[
   \frac{\partial}{\partial y}(\sqrt{xy}) = \frac{x}{2\sqrt{xy}}
   \]
   \[
   \frac{\partial}{\partial y}(z) = 0
   \]
   Por lo tanto:
   \[
   \frac{\partial w}{\partial y} = xz + \frac{x}{2\sqrt{xy}}
   \]

3. **\(\frac{\partial w}{\partial z}\)**:
   \[
   \frac{\partial}{\partial z}(xyz) = xy
   \]
   \[
   \frac{\partial}{\partial z}(\sqrt{xy}) = 0
   \]
   \[
   \frac{\partial}{\partial z}(z) = 1
   \]
   Por lo tanto:
   \[
   \frac{\partial w}{\partial z} = xy + 1
   \]

---

### d) \( w = \cos(z)e^{xyz} \)

1. **\(\frac{\partial w}{\partial x}\)**:
   \[
   \frac{\partial w}{\partial x} = \cos(z) \cdot yze^{xyz}
   \]

2. **\(\frac{\partial w}{\partial y}\)**:
   \[
   \frac{\partial w}{\partial y} = \cos(z) \cdot xze^{xyz}
   \]

3. **\(\frac{\partial w}{\partial z}\)**:
   Aplicamos la regla del producto:
   \[
   \frac{\partial w}{\partial z} = -\sin(z)e^{xyz} + \cos(z) \cdot xye^{xyz}
   \]
   Por lo tanto:
   \[
   \frac{\partial w}{\partial z} = -\sin(z)e^{xyz} + \cos(z) \cdot xye^{xyz}
   \]

---

### e) \( w = x^2y^5z^7 - x + y^3 - z^5 \)

1. **\(\frac{\partial w}{\partial x}\)**:
   \[
   \frac{\partial}{\partial x}(x^2y^5z^7) = 2xy^5z^7
   \]
   \[
   \frac{\partial}{\partial x}(-x) = -1
   \]
   \[
   \frac{\partial}{\partial x}(y^3 - z^5) = 0
   \]
   Por lo tanto:
   \[
   \frac{\partial w}{\partial x} = 2xy^5z^7 - 1
   \]

2. **\(\frac{\partial w}{\partial y}\)**:
   \[
   \frac{\partial}{\partial y}(x^2y^5z^7) = 5x^2y^4z^7
   \]
   \[
   \frac{\partial}{\partial y}(y^3) = 3y^2
   \]
   \[
   \frac{\partial}{\partial y}(-x - z^5) = 0
   \]
   Por lo tanto:
   \[
   \frac{\partial w}{\partial y} = 5x^2y^4z^7 + 3y^2
   \]

3. **\(\frac{\partial w}{\partial z}\)**:
   \[
   \frac{\partial}{\partial z}(x^2y^5z^7) = 7x^2y^5z^6
   \]
   \[
   \frac{\partial}{\partial z}(-z^5) = -5z^4
   \]
   Por lo tanto:
   \[
   \frac{\partial w}{\partial z} = 7x^2y^5z^6 - 5z^4
   \]

---

### f) \( w = \ln(xy + z^2) \)

1. **\(\frac{\partial w}{\partial x}\)** (utilizamos la regla de la cadena):
   \[
   \frac{\partial}{\partial x} \left( \ln(xy + z^2) \right) = \frac{y}{xy + z^2}
   \]

2. **\(\frac{\partial w}{\partial y}\)**:
   \[
   \frac{\partial}{\partial y} \left( \ln(xy + z^2) \right) = \frac{x}{xy + z^2}
   \]

3. **\(\frac{\partial w}{\partial z}\)**:
   \[
   \frac{\partial}{\partial z} \left( \ln(xy + z^2) \right) = \frac{2z}{xy + z^2}
   \]

---

¿Te gustaría una explicación más detallada de alguno de los apartados o tienes alguna duda?

### Preguntas relacionadas:
1. ¿Cómo se aplica la regla de la cadena en derivadas parciales?
2. ¿Qué diferencias existen entre la derivada parcial y la total?
3. ¿Qué papel juegan las funciones implícitas en las derivadas parciales?
4. ¿Cómo influye el uso de productos y cocientes en las derivadas parciales?
5. ¿Cómo interpretar derivadas parciales en gráficos tridimensionales?

### Tip:
Cuando calcules derivadas parciales, trata de fijar las otras variables como constantes para facilitar el proceso.

En cada caso encontrar las derivadas parciales ∂w , ∂w y ∂w con respecto a x, y, y z para las funciones dadas.

Discover how to calculate partial derivatives for functions involving multiple variables, including exponential, trigonometric, and logarithmic terms. This solution provides a step-by-step breakdown of each derivative using calculus concepts like the chain rule, suitable for university-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation