The second-order derivative of which of the following functions is 5^x?

The correct answer is the function represented in the fourth image:
$$\frac{5^x}{{\left(\ln \right(5\left)\right)}^2}$$

Step-by-Step Explanation:
We are given that the second-order derivative of a function is $5^x$. Let this function be $f\left(x\right)$. Thus, we have:
$$f^{\prime \prime }\left(x\right)=5^x$$
To find the function $f\left(x\right)$, we need to integrate $5^x$ twice with respect to $x$.

Recall the general integration formula for exponential functions of the form $a^x$:
$$\int a^{x}dx=\frac{a^x}{\ln \left(a\right)}+C$$

First Integration (Finding the first derivative, $f^{\prime }\left(x\right)$):
Integrating $f^{\prime \prime }\left(x\right)$ once gives:
$$f^{\prime }\left(x\right)=\int 5^{x}dx=\frac{5^x}{\ln \left(5\right)}$$
(Ignoring the constant of integration as we are matching the specific function forms in the choices).

Second Integration (Finding the original function, $f\left(x\right)$):
Integrating $f^{\prime }\left(x\right)$ gives:
$$f\left(x\right)=\int \frac{5^x}{\ln \left(5\right)}dx$$
Since $\frac{1}{\ln \left(5\right)}$ is a constant coefficient, we can factor it out of the integral:
$$f\left(x\right)=\frac{1}{\ln \left(5\right)}\int 5^{x}dx$$
Now we perform the integration:
$$f\left(x\right)=\frac{1}{\ln \left(5\right)}·\frac{5^x}{\ln \left(5\right)}=\frac{5^x}{{\left(\ln \right(5\left)\right)}^2}$$

Alternative Verification:
We can verify this by differentiating $f\left(x\right)=\frac{5^x}{{\left(\ln \right(5\left)\right)}^2}$ twice:
1. First derivative:
$$f^{\prime }\left(x\right)=\frac{d}{dx}\left[\frac{5^x}{{\left(\ln \right(5\left)\right)}^2}\right]=\frac{5^x\ln \left(5\right)}{{\left(\ln \right(5\left)\right)}^2}=\frac{5^x}{\ln \left(5\right)}$$
2. Second derivative:
$$f^{\prime \prime }\left(x\right)=\frac{d}{dx}\left[\frac{5^x}{\ln \left(5\right)}\right]=\frac{5^x\ln \left(5\right)}{\ln \left(5\right)}=5^x$$
This confirms that the second-order derivative of the function is indeed $5^x$.