Find `dy/dx` if :
`y=1-2((5x)/(3x+2))^(2)+((5x)/(3x+2))^(3)`

Answer

To find \(\frac{dy}{dx}\) for the function

\[ y = 1 - 2\left(\frac{5x}{3x+2}\right)^2 + \left(\frac{5x}{3x+2}\right)^3, \]

we will follow these steps:

### Step 1: Substitute \(v\)
Let

\[ v = \frac{5x}{3x + 2}. \]

Then, we can rewrite \(y\) in terms of \(v\):

\[ y = 1 - 2v^2 + v^3. \]

### Step 2: Differentiate \(y\) with respect to \(v\)
Now, we differentiate \(y\) with respect to \(v\):

\[ \frac{dy}{dv} = \frac{d}{dv}(1 - 2v^2 + v^3) = 0 - 4v + 3v^2 = 3v^2 - 4v. \]

### Step 3: Differentiate \(v\) with respect to \(x\)
Next, we need to find \(\frac{dv}{dx}\). We use the quotient rule for differentiation:

\[ v = \frac{u}{w} \quad \text{where } u = 5x \text{ and } w = 3x + 2. \]

Using the quotient rule:

\[ \frac{dv}{dx} = \frac{w \frac{du}{dx} - u \frac{dw}{dx}}{w^2}. \]

Calculating \(\frac{du}{dx}\) and \(\frac{dw}{dx}\):

- \(\frac{du}{dx} = 5\)
- \(\frac{dw}{dx} = 3\)

Now substituting these into the quotient rule:

\[ \frac{dv}{dx} = \frac{(3x + 2)(5) - (5x)(3)}{(3x + 2)^2} = \frac{15x + 10 - 15x}{(3x + 2)^2} = \frac{10}{(3x + 2)^2}. \]

### Step 4: Apply the chain rule
Now we can find \(\frac{dy}{dx}\) using the chain rule:

\[ \frac{dy}{dx} = \frac{dy}{dv} \cdot \frac{dv}{dx} = (3v^2 - 4v) \cdot \frac{10}{(3x + 2)^2}. \]

### Step 5: Substitute back \(v\)
Substituting back \(v = \frac{5x}{3x + 2}\):

\[ \frac{dy}{dx} = (3\left(\frac{5x}{3x + 2}\right)^2 - 4\left(\frac{5x}{3x + 2}\right)) \cdot \frac{10}{(3x + 2)^2}. \]

### Step 6: Simplify the expression
Now we simplify:

1. Calculate \(3\left(\frac{5x}{3x + 2}\right)^2 = \frac{75x^2}{(3x + 2)^2}\).
2. Calculate \(4\left(\frac{5x}{3x + 2}\right) = \frac{20x}{3x + 2}\).

Thus, we have:

\[ \frac{dy}{dx} = \left(\frac{75x^2}{(3x + 2)^2} - \frac{20x(3x + 2)}{(3x + 2)^2}\right) \cdot \frac{10}{(3x + 2)^2}. \]

Combine the fractions:

\[ \frac{dy}{dx} = \frac{75x^2 - 60x - 40}{(3x + 2)^4} \cdot 10. \]

### Final Result
Thus, the final expression for \(\frac{dy}{dx}\) is:

\[ \frac{dy}{dx} = \frac{10(75x^2 - 60x - 40)}{(3x + 2)^4}. \]