Question

Match List-I with List-II:

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

Choose the correct answer from the options given below.

• (A) - (I), (B) - (II), (C) - (III), (D) - (IV)
• (A) - (III), (B) - (IV), (C) - (II), (D) - (I)
• (A) - (I), (B) - (IV), (C) - (II), (D) - (III)
• (A) - (III), (B) - (II), (C) - (IV), (D) - (I)

Hint:

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.

Answer 1

The Correct Option is D

To match the functions in List-I with their derivatives in List-II, calculate the derivatives:

For (A) \(f(x) = \frac{5^x}{\log_e 5}\):

\(f'(x) = 5^x.\)

Thus, (A) matches with (III).

For (B) \(f(x) = \log_e 5\):

\(f'(x) = 0.\)

Thus, (B) matches with (IV).

For (C) \(f(x) = 5^x \log_e 5\):

\(f'(x) = 5^x \log_e 5.\)

Thus, (C) matches with (II).

For (D) \(f(x) = 5^x\):

\(f'(x) = 5^x (\log_e 5)^2.\)

Thus, (D) matches with (I).

Final Matching: (A) - (III), (B) - (IV), (C) - (II), (D) - (I)

Answer 2

To match the functions in List-I with their derivatives in List-II, we need to calculate the derivatives of each function:

For (A) \( f(x) = \frac{5^x}{\log_e 5} \):

The derivative of \( f(x) \) is:

\( f'(x) = 5^x \),

Thus, (A) matches with (III).

For (B) \( f(x) = \log_e 5 \):

The derivative of \( f(x) \) is:

\( f'(x) = 0 \),

Thus, (B) matches with (IV).

For (C) \( f(x) = 5^x \log_e 5 \):

The derivative of \( f(x) \) is:

\( f'(x) = 5^x \log_e 5 \),

Thus, (C) matches with (II).

For (D) \( f(x) = 5^x \):

The derivative of \( f(x) \) is:

\( f'(x) = 5^x (\log_e 5)^2 \),

Thus, (D) matches with (I).

Final Matching:

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)