(i) Let `y=((2+5x)^(2))/(x^(3) -1) ` then find dy/dx

Answer

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

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

we will use the quotient rule for differentiation. The quotient rule states that if you have a function \( y = \frac{u}{v} \), then

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \( u \) is the numerator and \( v \) is the denominator.

### Step 1: Identify \( u \) and \( v \)

Let:
- \( u = (2 + 5x)^2 \)
- \( v = x^3 - 1 \)

### Step 2: Differentiate \( u \) and \( v \)

**Differentiate \( u \)**:
Using the chain rule, we first expand \( u \):

\[ u = (2 + 5x)^2 = 4 + 20x + 25x^2 \]

Now, differentiate \( u \):

\[ \frac{du}{dx} = 0 + 20 + 50x = 50x + 20 \]

**Differentiate \( v \)**:
Now differentiate \( v \):

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

### Step 3: Apply the Quotient Rule

Now, we can apply the quotient rule:

\[ \frac{dy}{dx} = \frac{(x^3 - 1)(50x + 20) - (2 + 5x)^2(3x^2)}{(x^3 - 1)^2} \]

### Step 4: Simplify the expression

Now, we will simplify the numerator:

1. Expand \( (x^3 - 1)(50x + 20) \):

\[ = 50x^4 + 20x^3 - 50x - 20 \]

2. Expand \( (2 + 5x)^2(3x^2) \):

\[ (4 + 20x + 25x^2)(3x^2) = 12x^2 + 60x^3 + 75x^4 \]

3. Combine these results:

\[ \frac{dy}{dx} = \frac{(50x^4 + 20x^3 - 50x - 20) - (12x^2 + 60x^3 + 75x^4)}{(x^3 - 1)^2} \]

4. Combine like terms in the numerator:

\[ = \frac{(50x^4 - 75x^4) + (20x^3 - 60x^3) - 12x^2 - 50x - 20}{(x^3 - 1)^2} \]

\[ = \frac{-25x^4 - 40x^3 - 12x^2 - 50x - 20}{(x^3 - 1)^2} \]

### Final Result

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

\[ \frac{dy}{dx} = \frac{-25x^4 - 40x^3 - 12x^2 - 50x - 20}{(x^3 - 1)^2} \]