Differentiate the following w.r.t. x :
`(x-2) (x+3) (2x +5)`

Answer

To differentiate the expression \((x-2)(x+3)(2x+5)\) with respect to \(x\), we will follow these steps:

### Step 1: Expand the Expression
First, we need to expand the expression \((x-2)(x+3)(2x+5)\).

1. Multiply the first two factors:
\[ (x-2)(x+3) = x^2 + 3x - 2x - 6 = x^2 + x - 6 \]

2. Now, multiply the result with the third factor \((2x + 5)\):
\[ (x^2 + x - 6)(2x + 5) \]
Expanding this:
- \(x^2 \cdot 2x = 2x^3\)
- \(x^2 \cdot 5 = 5x^2\)
- \(x \cdot 2x = 2x^2\)
- \(x \cdot 5 = 5x\)
- \(-6 \cdot 2x = -12x\)
- \(-6 \cdot 5 = -30\)

Now, combine all these terms:
\[ 2x^3 + (5x^2 + 2x^2) + (5x - 12x) - 30 = 2x^3 + 7x^2 - 7x - 30 \]

### Step 2: Differentiate the Expanded Expression
Now, we differentiate the expanded expression \(2x^3 + 7x^2 - 7x - 30\) with respect to \(x\).

Using the power rule \(\frac{d}{dx}(x^n) = n \cdot x^{n-1}\):
1. Differentiate \(2x^3\):
\[ \frac{d}{dx}(2x^3) = 2 \cdot 3x^{3-1} = 6x^2 \]

2. Differentiate \(7x^2\):
\[ \frac{d}{dx}(7x^2) = 7 \cdot 2x^{2-1} = 14x \]

3. Differentiate \(-7x\):
\[ \frac{d}{dx}(-7x) = -7 \]

4. Differentiate \(-30\) (constant):
\[ \frac{d}{dx}(-30) = 0 \]

Now, combine these results:
\[ \frac{d}{dx}(2x^3 + 7x^2 - 7x - 30) = 6x^2 + 14x - 7 \]

### Final Result
Thus, the derivative of the expression \((x-2)(x+3)(2x+5)\) with respect to \(x\) is:
\[ \boxed{6x^2 + 14x - 7} \]