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The given problem is:

\[
y = 4u^2 + 5
\]
where 
\[
u = 7 - 2x^2
\]
We are tasked with finding \(\frac{dy}{dx}\).

This requires applying the chain rule, as \(y\) is a function of \(u\) and \(u\) is a function of \(x\). Using the chain rule:

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

### Step 1: Find \(\frac{dy}{du}\)
From the expression \(y = 4u^2 + 5\), we differentiate \(y\) with respect to \(u\):

\[
\frac{dy}{du} = \frac{d}{du}(4u^2 + 5) = 8u
\]

### Step 2: Find \(\frac{du}{dx}\)
Now, we differentiate \(u = 7 - 2x^2\) with respect to \(x\):

\[
\frac{du}{dx} = \frac{d}{dx}(7 - 2x^2) = -4x
\]

### Step 3: Apply the chain rule
Now, using the chain rule:

\[
\frac{dy}{dx} = 8u \cdot (-4x) = -32xu
\]

### Step 4: Substitute \(u = 7 - 2x^2\) into the result
Substituting the value of \(u\) into the expression:

\[
\frac{dy}{dx} = -32x(7 - 2x^2)
\]

### Step 5: Simplify
Expand the expression:

\[
\frac{dy}{dx} = -32x(7 - 2x^2) = -224x + 64x^3
\]

Thus, the derivative \(\frac{dy}{dx}\) is:

\[
\frac{dy}{dx} = 64x^3 - 224x
\]

Would you like further details or have any questions about the solution?

Here are 5 related questions:
1. How does the chain rule work in more complex functions?
2. What would the result be if \( u \) were a more complex function of \( x \)?
3. How do we approach problems where multiple variables are involved?
4. Can you derive higher-order derivatives from this expression?
5. How does implicit differentiation differ from the chain rule?

**Tip:** When solving composite function problems, always identify the inner and outer functions to apply the chain rule correctly.

Given y = 4u^2 + 5 where u = 7 - 2x^2. Find dy/dx.

Learn how to find the derivative of y = 4u^2 + 5 where u = 7 - 2x^2 using the chain rule. This problem illustrates the use of differentiation techniques for composite functions, a key concept in calculus. Ideal for students in higher-level mathematics courses.

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