differentiate `3x^(2) +2x -5 ` w.r.t. x

Answer

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

### Step 1: Identify the function
We start with the function:
\[ y = 3x^2 + 2x - 5 \]

### Step 2: Apply the differentiation rules
We will differentiate each term of the function separately using the power rule of differentiation, which states that if \( y = ax^n \), then \( \frac{dy}{dx} = n \cdot ax^{n-1} \).

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

### Step 4: Differentiate the second term \( 2x \)
Again using the power rule:
\[ \frac{d}{dx}(2x) = 1 \cdot 2x^{1-1} = 2 \]

### Step 5: Differentiate the constant term \( -5 \)
The derivative of a constant is zero:
\[ \frac{d}{dx}(-5) = 0 \]

### Step 6: Combine the results
Now we combine the derivatives of each term:
\[ \frac{dy}{dx} = 6x + 2 + 0 \]
Thus, we have:
\[ \frac{dy}{dx} = 6x + 2 \]

### Final Answer
The derivative of \( 3x^2 + 2x - 5 \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 6x + 2 \]

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