Find the derivative of `f(x) = 2 x^(3) + 3x^(2) + 5x + 9` at x = 2

Answer

To find the derivative of the function \( f(x) = 2x^3 + 3x^2 + 5x + 9 \) at \( x = 2 \), we will follow these steps:

### Step 1: Differentiate the function
We need to find the derivative \( f'(x) \) of the function \( f(x) \).

Using the power rule of differentiation:
- The derivative of \( 2x^3 \) is \( 6x^2 \) (using \( n \cdot a x^{n-1} \) where \( n=3 \) and \( a=2 \)).
- The derivative of \( 3x^2 \) is \( 6x \) (using \( n=2 \) and \( a=3 \)).
- The derivative of \( 5x \) is \( 5 \) (using \( n=1 \) and \( a=5 \)).
- The derivative of the constant \( 9 \) is \( 0 \).

Thus, the derivative of the function is:
\[ f'(x) = 6x^2 + 6x + 5 \]

### Step 2: Evaluate the derivative at \( x = 2 \)
Now we will substitute \( x = 2 \) into the derivative \( f'(x) \).

Calculating \( f'(2) \):
\[ f'(2) = 6(2^2) + 6(2) + 5 \]
Calculating each term:
- \( 2^2 = 4 \)
- \( 6(4) = 24 \)
- \( 6(2) = 12 \)

Now, substituting these values back into the equation:
\[ f'(2) = 24 + 12 + 5 \]
Adding these values together:
\[ f'(2) = 41 \]

### Final Answer
The derivative of \( f(x) \) at \( x = 2 \) is \( 41 \).

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