| List-I (Function) | List-II (Derivative w.r.t. x) | |
|---|---|---|
| (A) \( \frac{5^x}{\ln 5} \) | (I) \(5^x (\ln 5)^2\) | |
| (B) \(\ln 5\) | (II) \(5^x \ln 5\) | |
| (C) \(5^x \ln 5\) | (III) \(5^x\) | |
| (D) \(5^x\) | (IV) 0 | |
When calculating derivatives of exponential functions like \( 5^x \), remember that the derivative of \( a^x \) is \( a^x \ln a \). For constants such as \( \log_e 5 \), their derivative is zero. Applying the chain rule or constant rule will help simplify the calculations.
To match the functions in List-I with their derivatives in List-II, the derivatives are calculated as follows:
For (A) \(f(x) = \frac{5^x}{\log_e 5}\):
\(f'(x) = \frac{d}{dx} \left(\frac{5^x}{\log_e 5}\right) = \frac{1}{\log_e 5} \frac{d}{dx}(5^x) = \frac{1}{\log_e 5} (5^x \log_e 5) = 5^x.\)
Therefore, (A) matches with (III).
For (B) \(f(x) = \log_e 5\):
\(f'(x) = \frac{d}{dx}(\log_e 5) = 0\) (since \(\log_e 5\) is a constant).
Therefore, (B) matches with (IV).
For (C) \(f(x) = 5^x \log_e 5\):
\(f'(x) = \frac{d}{dx}(5^x \log_e 5) = \log_e 5 \frac{d}{dx}(5^x) = \log_e 5 (5^x \log_e 5) = 5^x (\log_e 5)^2.\)
Therefore, (C) matches with (I).
For (D) \(f(x) = 5^x\):
\(f'(x) = \frac{d}{dx}(5^x) = 5^x \log_e 5.\)
Therefore, (D) matches with (II).
Final Matching: (A) - (III), (B) - (IV), (C) - (I), (D) - (II)