If f(x)=` 2 sinx -3x^(4)+8`, then find f'(x) is

Options

A. `2 sin x-12 x^(3)`
B. `2 cos x-12 x^(3)`
C. `2 cos x+12 x^(3)`
D. `2 sin x+12x^(3)`

Answer

To find the derivative of the function \( f(x) = 2 \sin x - 3x^4 + 8 \), we will apply the rules of differentiation step by step.

### Step 1: Identify the components of the function
The function consists of three terms:
1. \( 2 \sin x \)
2. \( -3x^4 \)
3. \( 8 \)

### Step 2: Differentiate each term
We will differentiate each term separately using the following rules:
- The derivative of \( \sin x \) is \( \cos x \).
- The derivative of \( x^n \) is \( n x^{n-1} \).
- The derivative of a constant is \( 0 \).

#### Differentiating the first term:
\[ \frac{d}{dx}(2 \sin x) = 2 \cos x \]

#### Differentiating the second term:
\[ \frac{d}{dx}(-3x^4) = -3 \cdot 4x^{4-1} = -12x^3 \]

#### Differentiating the third term:
\[ \frac{d}{dx}(8) = 0 \]

### Step 3: Combine the derivatives
Now we combine the derivatives of the three terms:
\[ f'(x) = 2 \cos x - 12x^3 + 0 \]

Thus, the final result is:
\[ f'(x) = 2 \cos x - 12x^3 \]

### Final Answer:
\[ f'(x) = 2 \cos x - 12x^3 \]

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