https://file.aiquickdraw.com/mathbot/file/1726625299642gv54dbnk.jpg

Let's begin by analyzing the expression provided in the image.

The function given is:

\[
y = \left( \frac{x^2}{4} + 3x - \frac{3}{x} \right)^3
\]

We are asked to write it in the form \( y = f(u) \) and \( u = g(x) \), and then find \(\frac{dy}{dx}\) as a function of \(x\).

### Step 1: Expressing the function in the form \( y = f(u) \) and \( u = g(x) \)

We define:

\[
u = g(x) = \frac{x^2}{4} + 3x - \frac{3}{x}
\]

Then, the function \(y\) becomes:

\[
y = f(u) = u^3
\]

### Step 2: Differentiating \( y = f(u) \) and \( u = g(x) \)

We will use the chain rule to find \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\]

#### Finding \(\frac{dy}{du}\):
Since \( y = u^3 \),

\[
\frac{dy}{du} = 3u^2
\]

#### Finding \(\frac{du}{dx}\):
We need to differentiate \( u = \frac{x^2}{4} + 3x - \frac{3}{x} \) with respect to \(x\):

\[
\frac{du}{dx} = \frac{d}{dx} \left( \frac{x^2}{4} \right) + \frac{d}{dx} (3x) - \frac{d}{dx} \left( \frac{3}{x} \right)
\]

- \(\frac{d}{dx} \left( \frac{x^2}{4} \right) = \frac{2x}{4} = \frac{x}{2}\)
- \(\frac{d}{dx} (3x) = 3\)
- \(\frac{d}{dx} \left( \frac{3}{x} \right) = -\frac{3}{x^2}\)

Thus,

\[
\frac{du}{dx} = \frac{x}{2} + 3 + \frac{3}{x^2}
\]

### Step 3: Substituting everything to find \(\frac{dy}{dx}\)

Now, we substitute \(\frac{dy}{du} = 3u^2\) and \(\frac{du}{dx} = \frac{x}{2} + 3 + \frac{3}{x^2}\):

\[
\frac{dy}{dx} = 3u^2 \cdot \left( \frac{x}{2} + 3 + \frac{3}{x^2} \right)
\]

Finally, substitute \(u = \frac{x^2}{4} + 3x - \frac{3}{x}\) back into the equation:

\[
\frac{dy}{dx} = 3 \left( \frac{x^2}{4} + 3x - \frac{3}{x} \right)^2 \cdot \left( \frac{x}{2} + 3 + \frac{3}{x^2} \right)
\]

This is the derivative \(\frac{dy}{dx}\) as a function of \(x\).

Would you like further clarifications or details?

Here are five related questions to further your understanding:
1. How would the process change if the exponent in the original function was different?
2. What is the significance of using the chain rule in this context?
3. Can you identify other problems where expressing functions as \(y = f(u)\) and \(u = g(x)\) simplifies differentiation?
4. How would you compute higher-order derivatives for this function?
5. How can the quotient rule assist in finding \(\frac{du}{dx}\) for more complex fractions?

**Tip:** Always make sure to simplify terms before differentiating to avoid unnecessary complications!

Write the following function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. y = ((x^2 / 4) + 3x - (3/x))^3

Learn how to differentiate a composite function using the chain rule. The problem involves expressing the function y = ((x^2 / 4) + 3x - (3/x))^3 in the form y = f(u) and u = g(x), followed by finding dy/dx. This step-by-step solution includes differentiation techniques and the application of the chain rule.

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