Given, `f(x)=x^3-5x+2`. Then f'(2) equals

Options

A. `3x^2-5`
B. 2
C. 7
D. 8

Answer

To find \( f'(2) \) for the function \( f(x) = x^3 - 5x + 2 \), we will follow these steps:

### Step 1: Find the derivative of the function \( f(x) \)
The first step is to differentiate the function \( f(x) \) with respect to \( x \).

\[ f(x) = x^3 - 5x + 2 \]

Using the power rule of differentiation:

- The derivative of \( x^3 \) is \( 3x^2 \).
- The derivative of \( -5x \) is \( -5 \).
- The derivative of the constant \( 2 \) is \( 0 \).

Thus, the derivative \( f'(x) \) is:

\[ f'(x) = 3x^2 - 5 \]

### Step 2: Substitute \( x = 2 \) into the derivative
Next, we need to evaluate the derivative at \( x = 2 \):

\[ f'(2) = 3(2^2) - 5 \]

Calculating \( 2^2 \):

\[ 2^2 = 4 \]

Now substituting back into the equation:

\[ f'(2) = 3(4) - 5 \]

Calculating \( 3(4) \):

\[ 3(4) = 12 \]

So now we have:

\[ f'(2) = 12 - 5 \]

### Step 3: Simplify to find the final answer
Now, we simplify:

\[ f'(2) = 7 \]

Thus, the value of \( f'(2) \) is:

\[ \boxed{7} \]

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