If `g(t)=1-4t^(2)," find "g^(')(1)`.

Answer

To find \( g'(1) \) for the function \( g(t) = 1 - 4t^2 \), we will follow these steps:

### Step 1: Differentiate the function \( g(t) \)
We start with the function:
\[ g(t) = 1 - 4t^2 \]
To find \( g'(t) \), we differentiate \( g(t) \) with respect to \( t \).

### Step 2: Apply the differentiation rules
The derivative of a constant (1) is 0, and we apply the power rule for the term \( -4t^2 \):
\[ g'(t) = \frac{d}{dt}(1) - \frac{d}{dt}(4t^2) = 0 - 4 \cdot 2t^{2-1} \]
This simplifies to:
\[ g'(t) = -8t \]

### Step 3: Evaluate the derivative at \( t = 1 \)
Now, we need to find \( g'(1) \):
\[ g'(1) = -8 \cdot 1 = -8 \]

### Conclusion
Thus, the value of \( g'(1) \) is:
\[ \boxed{-8} \]

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