We are given the function $z = \sqrt{1 - x + 4xy^2}$, where $x = \ln(t)$ and $y = t^{3/2}$, and we need to find $\frac{dz}{dt}$ using the chain rule.

Step 1: Differentiate $z$ with respect to $t$ using the Chain Rule

The Chain Rule tells us: $\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}$

Step 2: Compute the partial derivatives of $z$

First, rewrite $z$ for clarity: $z = (1 - x + 4xy^2)^{1/2}$

• The partial derivative of $z$ with respect to $x$: $\frac{\partial z}{\partial x} = \frac{1}{2}(1 - x + 4xy^2)^{-1/2}(-1 + 4y^2)$

• The partial derivative of $z$ with respect to $y$: $\frac{\partial z}{\partial y} = \frac{1}{2}(1 - x + 4xy^2)^{-1/2}(8xy)$

Step 3: Compute the derivatives of $x$ and $y$ with respect to $t$

• The derivative of $x = \ln(t)$ with respect to $t$: $\frac{dx}{dt} = \frac{1}{t}$

• The derivative of $y = t^{3/2}$ with respect to $t$: $\frac{dy}{dt} = \frac{3}{2}t^{1/2}$

Step 4: Combine everything

Now substitute these derivatives into the Chain Rule formula: [ \frac{dz}{dt} = \left[\frac{1}{2}(1 - x + 4xy^2)^{-1/2}(-1 + 4y^2)\right] \cdot \frac{1}{t} + \left[\frac{1}{2}(1 - x + 4xy^2)^{-1/2}(8xy)\right] \cdot \frac{3}{2}t^{1/2} ]

This is the expression for $\frac{dz}{dt}$, which can be simplified further, but it is now set up in terms of $t$.

Would you like me to simplify this further or explain any part in detail?

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Here are 5 related questions to further your understanding:

1. How does the chain rule apply to functions of several variables?
2. What is the physical significance of partial derivatives in multivariable calculus?
3. Can we use implicit differentiation in this problem, and how would it affect the result?
4. How does logarithmic differentiation help in solving derivatives involving logarithmic functions?
5. How can this problem be extended to higher-order derivatives?

Tip: When solving problems involving multiple variables, always make sure to correctly apply the chain rule, carefully differentiating each variable and considering how they depend on each other.