Problem Analysis
We are given that \( u = f(x, y, z) \), where \( x = f(t) \), \( y = f(t) \), and \( z = f(t) \). We need to find the derivative of \( u \) with respect to \( t \), i.e., \( \frac{du}{dt} \).
This is a classic application of the
chain rule for multivariable functions. The chain rule allows us to compute the derivative of a composite function by considering how each variable depends on the parameter \( t \).
Step-by-Step Solution
1.
Express \( u \) as a function of \( t \):
Since \( u = f(x, y, z) \) and \( x = f(t) \), \( y = f(t) \), and \( z = f(t) \), we can write:
\[
u = f(x(t), y(t), z(t))
\]
2.
Apply the chain rule:
The chain rule for a function of multiple variables states:
\[
\frac{du}{dt} = \frac{\partial u}{\partial x} \frac{dx}{dt} + \frac{\partial u}{\partial y} \frac{dy}{dt} + \frac{\partial u}{\partial z} \frac{dz}{dt}
\]
Here:
- \( \frac{\partial u}{\partial x} \) is the partial derivative of \( u \) with respect to \( x \).
- \( \frac{\partial u}{\partial y} \) is the partial derivative of \( u \) with respect to \( y \).
- \( \frac{\partial u}{\partial z} \) is the partial derivative of \( u \) with respect to \( z \).
- \( \frac{dx}{dt} \), \( \frac{dy}{dt} \), and \( \frac{dz}{dt} \) are the derivatives of \( x \), \( y \), and \( z \) with respect to \( t \), respectively.
3.
Substitute the partial derivatives and derivatives:
Using the notation \( \frac{\partial u}{\partial x} = \frac{\partial f}{\partial x} \), \( \frac{\partial u}{\partial y} = \frac{\partial f}{\partial y} \), and \( \frac{\partial u}{\partial z} = \frac{\partial f}{\partial z} \), we can rewrite the chain rule as:
\[
\frac{du}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial z} \frac{dz}{dt}
\]
4.
Match with the given options:
Comparing this result with the provided options, we see that it matches option (D):
\[
\frac{du}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial z} \frac{dz}{dt}
\]
Final Answer
\[
\boxed{D}
\]
Parent Tip: Review the logic above to help your child master the concept of gre math worksheet.
