For each of the following fuctions, evaluate the derivative at the indicated value (s) :
`g(x)=4x^(8), x=-1/2, x=1/2`

Answer

To find the derivative of the function \( g(x) = 4x^8 \) at the specified values \( x = -\frac{1}{2} \) and \( x = \frac{1}{2} \), we will follow these steps:

### Step 1: Find the derivative of \( g(x) \)

The function is given as:
\[ g(x) = 4x^8 \]

To find the derivative \( g'(x) \), we apply the power rule, which states that if \( f(x) = ax^n \), then \( f'(x) = n \cdot ax^{n-1} \).

Applying the power rule:
\[ g'(x) = 8 \cdot 4x^{8-1} = 32x^7 \]

### Step 2: Evaluate the derivative at \( x = -\frac{1}{2} \)

Now we will substitute \( x = -\frac{1}{2} \) into the derivative:
\[ g'\left(-\frac{1}{2}\right) = 32\left(-\frac{1}{2}\right)^7 \]

Calculating \( \left(-\frac{1}{2}\right)^7 \):
\[ \left(-\frac{1}{2}\right)^7 = -\frac{1}{128} \]

Now substituting this back into the derivative:
\[ g'\left(-\frac{1}{2}\right) = 32 \cdot \left(-\frac{1}{128}\right) = -\frac{32}{128} = -\frac{1}{4} \]

### Step 3: Evaluate the derivative at \( x = \frac{1}{2} \)

Next, we will substitute \( x = \frac{1}{2} \) into the derivative:
\[ g'\left(\frac{1}{2}\right) = 32\left(\frac{1}{2}\right)^7 \]

Calculating \( \left(\frac{1}{2}\right)^7 \):
\[ \left(\frac{1}{2}\right)^7 = \frac{1}{128} \]

Now substituting this back into the derivative:
\[ g'\left(\frac{1}{2}\right) = 32 \cdot \frac{1}{128} = \frac{32}{128} = \frac{1}{4} \]

### Final Answers:
- \( g'\left(-\frac{1}{2}\right) = -\frac{1}{4} \)
- \( g'\left(\frac{1}{2}\right) = \frac{1}{4} \)

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