When finding the second derivative of exponential functions like \( 5^x \), it is important to apply the chain rule properly. For each option, carefully differentiate, considering constants and the behavior of exponential functions. Pay special attention to terms like \( \ln(5) \), which will affect the derivatives but not change the fundamental behavior of the exponential function itself. If the second derivative doesn't match the desired form, eliminate that option.
We need to determine which function’s second derivative equals \(5^x\). Let us check each option.
For (1) : \(5^x \ln(5)\):
\[ \frac{d}{dx} \left(5^x \ln(5)\right) = 5^x (\ln(5))^2 \]
\[ \frac{d^2}{dx^2} \left(5^x \ln(5)\right) = 5^x (\ln(5))^3 \neq 5^x \]
For (2) : \(5^x (\ln(5))^2\):
\[ \frac{d}{dx} \left(5^x (\ln(5))^2\right) = 5^x (\ln(5))^3 \]
\[ \frac{d^2}{dx^2} \left(5^x (\ln(5))^2\right) = 5^x (\ln(5))^4 \neq 5^x \]
For (3) : \(\frac{5^x}{\ln(5)}\):
\[ \frac{d}{dx} \left(\frac{5^x}{\ln(5)}\right) = 5^x \]
\[ \frac{d^2}{dx^2} \left(\frac{5^x}{\ln(5)}\right) = 5^x \ln(5) \neq 5^x \]
For (4) : \(\frac{5^x}{(\ln(5))^2}\):
\[ \frac{d}{dx} \left(\frac{5^x}{(\ln(5))^2}\right) = \frac{5^x \ln(5)}{(\ln(5))^2} \]
\[ \frac{d^2}{dx^2} \left(\frac{5^x}{(\ln(5))^2}\right) = 5^x \]
Thus, the correct answer is:
\[ \frac{5^x}{(\ln(5))^2} \]
We need to determine which function’s second derivative equals \( 5^x \). Let us check each option:
For (1) : \( 5^x \ln(5) \):
First, calculate the first derivative:
\[ \frac{d}{dx} \left(5^x \ln(5)\right) = 5^x (\ln(5))^2 \] Now, calculate the second derivative: \[ \frac{d^2}{dx^2} \left(5^x \ln(5)\right) = 5^x (\ln(5))^3 \neq 5^x \] Thus, option (1) is not the correct answer.For (2) : \( 5^x (\ln(5))^2 \):
First, calculate the first derivative: \[ \frac{d}{dx} \left(5^x (\ln(5))^2\right) = 5^x (\ln(5))^3 \] Now, calculate the second derivative: \[ \frac{d^2}{dx^2} \left(5^x (\ln(5))^2\right) = 5^x (\ln(5))^4 \neq 5^x \] Thus, option (2) is not the correct answer.For (3) : \( \frac{5^x}{\ln(5)} \):
First, calculate the first derivative: \[ \frac{d}{dx} \left(\frac{5^x}{\ln(5)}\right) = 5^x \] Now, calculate the second derivative: \[ \frac{d^2}{dx^2} \left(\frac{5^x}{\ln(5)}\right) = 5^x \ln(5) \neq 5^x \] Thus, option (3) is not the correct answer.For (4) : \( \frac{5^x}{(\ln(5))^2} \):
First, calculate the first derivative: \[ \frac{d}{dx} \left(\frac{5^x}{(\ln(5))^2}\right) = \frac{5^x \ln(5)}{(\ln(5))^2} \] Now, calculate the second derivative: \[ \frac{d^2}{dx^2} \left(\frac{5^x}{(\ln(5))^2}\right) = 5^x \] Thus, the correct answer is option (4).Conclusion: The correct answer is:
\[ \frac{5^x}{(\ln(5))^2} \]List-I (Name of account to be debited or credited, when shares are forfeited) | List-II (Amount to be debited or credited) |
---|---|
(A) Share Capital Account | (I) Debited with amount not received |
(B) Share Forfeited Account | (II) Credited with amount not received |
(C) Calls-in-arrears Account | (III) Credited with amount received towards share capital |
(D) Securities Premium Account | (IV) Debited with amount called up |