Step 1: Understanding the Concept:
To find the derivative of a piecewise function at a specific point, we use the rule defined for the interval containing that point. Since we need \( f'(-1) \) and \( -1<0 \), we only consider the piece \( f(x) = 4(5^x) \)[cite: 1].
Step 2: Key Formula or Approach:
The derivative of an exponential function \( a^x \) is \( \frac{d}{dx}(a^x) = a^x \log a \) (or \( a^x \ln a \))[cite: 1].
Step 3: Detailed Explanation:
For \( x<0 \), \( f(x) = 4 \cdot 5^x \).
Differentiating with respect to \( x \):
\[ f'(x) = 4 \cdot (5^x \log 5) \]
Now, substitute \( x = -1 \):
\[ f'(-1) = 4 \cdot (5^{-1} \log 5) \]
\[ f'(-1) = 4 \cdot \frac{1}{5} \log 5 = \frac{4}{5} \log 5 \]
Step 4: Final Answer:
The value of \( f'(-1) \) is \( \frac{4}{5} \log 5 \).