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NDA 2025
Exam
NDA
Year
2025
Subject
Mathematics
Chapter
Differentiation

NDA 2025 · Mathematics · Differentiation

For the following two (02) items: Consider the function f(x) =...

For the following two (02) items:
Consider the function f(x)=f(x) =\begin{cases} 4(5^x), & x < 0 \\ 8k + x, & x \geq 0 \end{cases}$.<br><br>Whatis.<br><br>What isf'(-1)$ equal to?

Options

  1. A25ln⁡5\dfrac{2}{5}\ln 5
  2. B35ln⁡5\dfrac{3}{5}\ln 5
  3. C45ln⁡5\dfrac{4}{5}\ln 5
  4. D20ln⁡520\ln 5

Correct answer

C. 45ln⁡5\dfrac{4}{5}\ln 5

Step-by-step solution

For x<0x < 0, the function is given by f(x)=4(5x)f(x) = 4(5^x).

Differentiating with respect to xx, we get:

f′(x)=4(5xln⁡5)f'(x) = 4(5^x \ln 5)

To find f′(−1)f'(-1), substitute x=−1x = -1 into the derivative:

f′(−1)=4(5−1ln⁡5)f'(-1) = 4(5^{-1} \ln 5)

f′(−1)=45ln⁡5f'(-1) = \dfrac{4}{5}\ln 5

Answer: 45ln⁡5\dfrac{4}{5}\ln 5
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