Let `G(x)=2-x^(3)+x^(5)/5`. Find `G^(')(-2)`.

Answer

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

### Step 1: Differentiate \( G(x) \)

We start by differentiating \( G(x) \) with respect to \( x \).

\[ G(x) = 2 - x^3 + \frac{x^5}{5} \]

Using the rules of differentiation:

1. The derivative of a constant (2) is 0.
2. The derivative of \( -x^3 \) is \( -3x^2 \).
3. The derivative of \( \frac{x^5}{5} \) is \( \frac{5x^4}{5} = x^4 \).

Putting it all together:

\[ G'(x) = 0 - 3x^2 + x^4 \]

Thus, we have:

\[ G'(x) = x^4 - 3x^2 \]

### Step 2: Evaluate \( G'(-2) \)

Now we need to evaluate \( G'(-2) \):

\[ G'(-2) = (-2)^4 - 3(-2)^2 \]

Calculating each term:

1. \( (-2)^4 = 16 \)
2. \( (-2)^2 = 4 \) and \( 3 \times 4 = 12 \)

Now substituting back into the equation:

\[ G'(-2) = 16 - 12 \]

Calculating the final result:

\[ G'(-2) = 4 \]

### Final Answer

Thus, the value of \( G'(-2) \) is \( 4 \).

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