Use the Chain Rule to find dz/dt
. Where:

z=cos(x+8y),x=t4,y=3t

∂z/∂x=
 
functionsequation editor

dx/dt=
 
functionsequation editor

∂z/∂y=
 
functionsequation editor

dy/dt=
 
functionsequation editor
dz/dt=
 
functionsequation editor

To find \(\frac{dz}{dt}\) using the **Chain Rule**, we are given:

\[
z = \cos(x + 8y), \quad x = t^4, \quad y = 3t
\]

### Step 1: Find the partial derivatives

1. **\(\frac{\partial z}{\partial x}\)**:
\[
\frac{\partial z}{\partial x} = \frac{d}{dx}[\cos(x + 8y)] = -\sin(x + 8y)
\]

2. **\(\frac{dx}{dt}\)**:
\[
\frac{dx}{dt} = \frac{d}{dt}[t^4] = 4t^3
\]

3. **\(\frac{\partial z}{\partial y}\)**:
\[
\frac{\partial z}{\partial y} = \frac{d}{dy}[\cos(x + 8y)] = -\sin(x + 8y) \cdot 8 = -8\sin(x + 8y)
\]

4. **\(\frac{dy}{dt}\)**:
\[
\frac{dy}{dt} = \frac{d}{dt}[3t] = 3
\]

### Step 2: Apply the Chain Rule

The Chain Rule for finding \(\frac{dz}{dt}\) is:

\[
\frac{dz}{dt} = \frac{\partial z}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial z}{\partial y} \cdot \frac{dy}{dt}
\]

Substituting the partial derivatives and rates of change:

\[
\frac{dz}{dt} = -\sin(x + 8y) \cdot 4t^3 + (-8\sin(x + 8y)) \cdot 3
\]

\[
\frac{dz}{dt} = -4t^3\sin(x + 8y) - 24\sin(x + 8y)
\]

\[
\frac{dz}{dt} = -\sin(x + 8y)(4t^3 + 24)
\]

So, the final expression is:

\[
\boxed{\frac{dz}{dt} = -\sin(x + 8y)(4t^3 + 24)}
\]

Let me know if you'd like more details or have any questions!

---

### Related Questions:
1. What if \(x = t^2\) instead of \(t^4\)? How would the result change?
2. How does the Chain Rule apply when multiple variables are dependent on \(t\)?
3. Can you find \(\frac{dz}{dt}\) for different trigonometric functions like \(\sin\) or \(\tan\)?
4. What happens if \(y = 2t\) instead of \(y = 3t\)?
5. How can we apply the product rule along with the chain rule?

### Tip:
Always take care to identify dependent variables and apply the chain rule correctly for each one when multiple variables depend on \(t\).

Use the Chain Rule to find dz/dt where z = cos(x + 8y), x = t^4, y = 3t.

Learn how to use the Chain Rule to find dz/dt for the function z = cos(x + 8y), where x = t^4 and y = 3t. This problem involves partial derivatives and trigonometric functions, making it perfect for undergraduate calculus 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