Find `dy/dx` if :
`y=9u^(2),u=1-3/2x^(2)`

Answer

To find \(\frac{dy}{dx}\) given \(y = 9u^2\) and \(u = 1 - \frac{3}{2}x^2\), we will use the chain rule of differentiation.

### Step 1: Differentiate \(y\) with respect to \(u\)

Given:
\[ y = 9u^2 \]

Differentiating \(y\) with respect to \(u\):
\[ \frac{dy}{du} = 18u \]

### Step 2: Differentiate \(u\) with respect to \(x\)

Given:
\[ u = 1 - \frac{3}{2}x^2 \]

Differentiating \(u\) with respect to \(x\):
\[ \frac{du}{dx} = 0 - 3x = -3x \]

### Step 3: Apply the chain rule to find \(\frac{dy}{dx}\)

Using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

Substituting the values we found:
\[ \frac{dy}{dx} = 18u \cdot (-3x) = -54ux \]

### Step 4: Substitute \(u\) back into the equation

Now, substitute \(u = 1 - \frac{3}{2}x^2\) into the equation:
\[ \frac{dy}{dx} = -54 \left(1 - \frac{3}{2}x^2\right)x \]

### Final Expression

Thus, the final expression for \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -54x \left(1 - \frac{3}{2}x^2\right) \]