Understanding the Second Derivative of Functions

The question asks us to identify which of the given functions has a second order derivative equal to \( 5^x \). The second order derivative of a function is found by differentiating the function twice. We will examine each option by calculating its first and second derivatives.

Recall the basic rules of differentiation for exponential functions:

• The derivative of \( a^x \) with respect to \( x \) is \( \frac{d}{dx}(a^x) = a^x \ln a \), where \( \ln a \) is the natural logarithm of \( a \).
• The derivative of \( c \cdot f(x) \) with respect to \( x \) is \( c \cdot f'(x) \), where \( c \) is a constant.

Calculating Derivatives for Each Option

Option 1: \( f(x) = 5^x \log_e 5 \)

Here, \( \log_e 5 \) is a constant. Let's find the first derivative:

\( f'(x) = \frac{d}{dx}(5^x \log_e 5) \)

Using the constant multiple rule and the derivative of \( a^x \):

\( f'(x) = (\log_e 5) \cdot \frac{d}{dx}(5^x) \) \( f'(x) = (\log_e 5) \cdot (5^x \log_e 5) \) \( f'(x) = 5^x (\log_e 5)^2 \)

Now, let's find the second derivative:

\( f''(x) = \frac{d}{dx}(5^x (\log_e 5)^2) \)

Again, \( (\log_e 5)^2 \) is a constant:

\( f''(x) = (\log_e 5)^2 \cdot \frac{d}{dx}(5^x) \) \( f''(x) = (\log_e 5)^2 \cdot (5^x \log_e 5) \) \( f''(x) = 5^x (\log_e 5)^3 \)

This is not equal to \( 5^x \).

Option 2: \( f(x) = 5^x (\log_e 5)^2 \)

Here, \( (\log_e 5)^2 \) is a constant. Let's find the first derivative:

\( f'(x) = \frac{d}{dx}(5^x (\log_e 5)^2) \)

Using the constant multiple rule and the derivative of \( a^x \):

\( f'(x) = (\log_e 5)^2 \cdot \frac{d}{dx}(5^x) \) \( f'(x) = (\log_e 5)^2 \cdot (5^x \log_e 5) \) \( f'(x) = 5^x (\log_e 5)^3 \)

Now, let's find the second derivative:

\( f''(x) = \frac{d}{dx}(5^x (\log_e 5)^3) \)

Again, \( (\log_e 5)^3 \) is a constant:

\( f''(x) = (\log_e 5)^3 \cdot \frac{d}{dx}(5^x) \) \( f''(x) = (\log_e 5)^3 \cdot (5^x \log_e 5) \) \( f''(x) = 5^x (\log_e 5)^4 \)

This is not equal to \( 5^x \).

Option 3: \( f(x) = 5^x / \log_e 5 \)

We can write this as \( f(x) = \frac{1}{\log_e 5} \cdot 5^x \). Here, \( \frac{1}{\log_e 5} \) is a constant. Let's find the first derivative:

\( f'(x) = \frac{d}{dx}\left(\frac{1}{\log_e 5} \cdot 5^x\right) \)

Using the constant multiple rule and the derivative of \( a^x \):

\( f'(x) = \frac{1}{\log_e 5} \cdot \frac{d}{dx}(5^x) \) \( f'(x) = \frac{1}{\log_e 5} \cdot (5^x \log_e 5) \) \( f'(x) = 5^x \)

Now, let's find the second derivative:

\( f''(x) = \frac{d}{dx}(5^x) \) \( f''(x) = 5^x \log_e 5 \)

This is not equal to \( 5^x \).

Option 4: \( f(x) = 5^x / (\log_e 5)^2 \)

We can write this as \( f(x) = \frac{1}{(\log_e 5)^2} \cdot 5^x \). Here, \( \frac{1}{(\log_e 5)^2} \) is a constant. Let's find the first derivative:

\( f'(x) = \frac{d}{dx}\left(\frac{1}{(\log_e 5)^2} \cdot 5^x\right) \)

Using the constant multiple rule and the derivative of \( a^x \):

\( f'(x) = \frac{1}{(\log_e 5)^2} \cdot \frac{d}{dx}(5^x) \) \( f'(x) = \frac{1}{(\log_e 5)^2} \cdot (5^x \log_e 5) \) \( f'(x) = \frac{5^x \log_e 5}{(\log_e 5)^2} \) \( f'(x) = \frac{5^x}{\log_e 5} \)

Now, let's find the second derivative:

\( f''(x) = \frac{d}{dx}\left(\frac{5^x}{\log_e 5}\right) \)

We can write this as \( f'(x) = \frac{1}{\log_e 5} \cdot 5^x \). Again, \( \frac{1}{\log_e 5} \) is a constant.

\( f''(x) = \frac{1}{\log_e 5} \cdot \frac{d}{dx}(5^x) \) \( f''(x) = \frac{1}{\log_e 5} \cdot (5^x \log_e 5) \) \( f''(x) = 5^x \)

The second derivative of this function is indeed \( 5^x \).

Summary of Second Derivatives

From the table, we can see that the function \( f(x) = 5^x / (\log_e 5)^2 \) has a second derivative of \( 5^x \).

Alternative Approach: Integration

Another way to solve this problem is by integrating the given second derivative \( 5^x \) twice.

Let \( f''(x) = 5^x \). To find \( f'(x) \), we integrate \( f''(x) \) with respect to \( x \):

\( f'(x) = \int 5^x dx \)

Using the integration rule \( \int a^x dx = \frac{a^x}{\ln a} + C \):

\( f'(x) = \frac{5^x}{\ln 5} + C_1 \)

Where \( C_1 \) is the constant of integration. Now, to find \( f(x) \), we integrate \( f'(x) \) with respect to \( x \):

\( f(x) = \int \left(\frac{5^x}{\ln 5} + C_1\right) dx \) \( f(x) = \int \frac{5^x}{\ln 5} dx + \int C_1 dx \)

Treating \( \frac{1}{\ln 5} \) and \( C_1 \) as constants:

\( f(x) = \frac{1}{\ln 5} \int 5^x dx + C_1 \int dx \) \( f(x) = \frac{1}{\ln 5} \left(\frac{5^x}{\ln 5}\right) + C_1 x + C_2 \) \( f(x) = \frac{5^x}{(\ln 5)^2} + C_1 x + C_2 \)

Where \( C_2 \) is another constant of integration. The question asks for *a* function whose second derivative is \( 5^x \). This means we can choose the simplest form where \( C_1 = 0 \) and \( C_2 = 0 \). In that case, the function is:

\( f(x) = \frac{5^x}{(\ln 5)^2} \)

Since \( \ln 5 = \log_e 5 \), this is \( f(x) = \frac{5^x}{(\log_e 5)^2} \), which matches Option 4.

Conclusion

By calculating the second derivative of each given function or by integrating the target second derivative twice, we find that the function \( 5^x / (\log_e 5)^2 \) is the one whose second order derivative is \( 5^x \).

Revision Table: Key Differentiation and Integration Formulas

Additional Information: Order of Derivatives and Integration

The order of a derivative indicates how many times a function has been differentiated. The first derivative (\( f'(x) \) or \( \frac{dy}{dx} \)) gives the rate of change of the function. The second derivative (\( f''(x) \) or \( \frac{d^2y}{dx^2} \)) gives the rate of change of the first derivative, often related to the concavity of the function's graph. Higher-order derivatives can be found by differentiating repeatedly.

Integration is the reverse process of differentiation. If we know the derivative of a function, we can integrate it to find the original function. When performing indefinite integration (without limits), we must always add a constant of integration (like \( C_1 \) or \( C_2 \)). This is because the derivative of a constant is zero, so any constant term in the original function is lost during differentiation. When we integrate, we need to account for this potential constant. To find the function from its second derivative, we perform two integrations, which introduce two constants of integration. In this specific problem, the options do not include these arbitrary constants, suggesting we are looking for a specific antiderivative (the one where constants are zero).

Understanding the relationship between differentiation and integration is fundamental in calculus for solving various problems involving rates of change, areas, volumes, and more complex applications.

Let’s differentiate the general exponential function: \[ \frac{d}{dx}(a^x) = a^x \ln a \] So, \[ \frac{d}{dx}(5^x) = 5^x \ln 5 \quad \text{and} \quad \frac{d^2}{dx^2}(5^x) = (5^x \ln 5)' = 5^x (\ln 5)^2 \]

Therefore, to get: \[ \frac{d^2}{dx^2}(f(x)) = 5^x \] We must have: \[ f(x) = \frac{5^x}{(\ln 5)^2} \]

✅ Now let's verify: 
Let \( f(x) = \frac{5^x}{(\ln 5)^2} \) 
First derivative: \[ f'(x) = \frac{5^x \ln 5}{(\ln 5)^2} = \frac{5^x}{\ln 5} \] Second derivative: \[ f''(x) = \frac{5^x \ln 5}{\ln 5} = 5^x \]

Answer:

\( \frac{5^x}{(\log_e 5)^2} \)

Step 1: Understand the Problem

We need to find which function's second derivative equals 5^x.

Step 2: Recall Derivative Rules for Exponential Functions

For a function \( f(x) = a^x \): \[ f'(x) = a^x \ln a \] \[ f''(x) = a^x (\ln a)^2 \]

Step 3: Apply to Our Case (a = 5)

If we want \( f''(x) = 5^x \), then: \[ 5^x (\ln 5)^2 = 5^x \] This would require \( (\ln 5)^2 = 1 \), which is false.

Step 4: Consider Alternative Approach

We need a function whose second derivative is 5^x. Let's find \( f(x) \) such that: \[ \frac{d^2}{dx^2} f(x) = 5^x \]

Step 5: Integrate Twice

First integration: \[ f'(x) = \int 5^x dx = \frac{5^x}{\ln 5} + C_1 \] Second integration: \[ f(x) = \int \frac{5^x}{\ln 5} dx = \frac{5^x}{(\ln 5)^2} + C_1x + C_2 \] The simplest solution (ignoring constants) is: \[ f(x) = \frac{5^x}{(\ln 5)^2} \]

Step 6: Verify the Solution

Differentiate \( f(x) = \frac{5^x}{(\ln 5)^2} \) twice: \[ f'(x) = \frac{5^x \ln 5}{(\ln 5)^2} = \frac{5^x}{\ln 5} \] \[ f''(x) = \frac{5^x \ln 5}{\ln 5} = 5^x \] This matches our requirement.

Final Answer:

The function whose second derivative is 5^x is \[ \boxed{\frac{5^x}{(\log_e 5)^2}} \].