The derivative of `2x^(3)-3x^(2)-5x+6` at `x=1` is

Options

A. A. 0
B. B. `-6`
C. C. `-5`
D. D. 5

Answer

To find the derivative of the function \( f(x) = 2x^3 - 3x^2 - 5x + 6 \) at \( x = 1 \), 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, which states that if \( f(x) = ax^n \), then \( f'(x) = n \cdot ax^{n-1} \):

1. The derivative of \( 2x^3 \) is \( 6x^2 \) (since \( 3 \cdot 2 = 6 \) and \( x^{3-1} = x^2 \)).
2. The derivative of \( -3x^2 \) is \( -6x \) (since \( 2 \cdot -3 = -6 \) and \( x^{2-1} = x \)).
3. The derivative of \( -5x \) is \( -5 \) (since the derivative of \( x \) is \( 1 \)).
4. The derivative of the constant \( 6 \) is \( 0 \).

So, combining these results, we have:
\[ f'(x) = 6x^2 - 6x - 5 \]

### Step 2: Evaluate the derivative at \( x = 1 \)
Now we need to find \( f'(1) \):
\[ f'(1) = 6(1)^2 - 6(1) - 5 \]

Calculating this step-by-step:
1. \( 6(1)^2 = 6 \)
2. \( -6(1) = -6 \)
3. So, \( f'(1) = 6 - 6 - 5 \)

Now, simplifying:
\[ f'(1) = 6 - 6 - 5 = -5 \]

### Final Answer
The derivative of \( 2x^3 - 3x^2 - 5x + 6 \) at \( x = 1 \) is \( -5 \).

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