Question

If $f(x) = \begin{cases} 4(5^{x}) & x < 0 \\ 8k + x & x \ge 0 \end{cases}$ then $f'(-1) =$

• A. $\frac{2}{5}\log 5$
• B. $\frac{4}{5}\log 5$
• C. $\frac{3}{5}\log 5$
• D. $20\log 5$

Hint:

Always identify which "piece" of a piecewise function your point falls into before differentiating.

Answer 1

The Correct Option is B

Step 1: Concept
To find $f'(-1)$, we must use the piece of the function defined for the interval containing $-1$, which is $x < 0$.

Step 2: Meaning
The derivative of an exponential function $a^x$ is $a^x \log a$.

Step 3: Analysis
For $x < 0$, $f(x) = 4(5^x)$. Differentiating with respect to $x$ gives $f'(x) = 4(5^x \log 5)$.

Step 4: Conclusion
Substitute $x = -1$: $f'(-1) = 4(5^{-1} \log 5) = 4(\frac{1}{5} \log 5) = \frac{4}{5} \log 5$.
Final Answer: (B)