1. The problem is to find the derivatives of logarithmic and exponential functions. 2. The derivative of the natural logarithm function $\ln(x)$ is given by the formula: $$\frac{d}{dx} \ln(x) = \frac{1}{x}$$ This is valid for $x > 0$. 3. For exponential functions of the form $a^x$ where $a > 0$ and $a \neq 1$, the derivative is: $$\frac{d}{dx} a^x = a^x \ln(a)$$ 4. For the natural exponential function $e^x$, the derivative is: $$\frac{d}{dx} e^x = e^x$$ 5. Example: Find the derivative of $f(x) = \ln(x^2 + 1)$. 6. Using the chain rule: $$f'(x) = \frac{1}{x^2 + 1} \cdot \frac{d}{dx} (x^2 + 1)$$ 7. Compute the inner derivative: $$\frac{d}{dx} (x^2 + 1) = 2x$$ 8. Substitute back: $$f'(x) = \frac{1}{x^2 + 1} \cdot 2x = \frac{2x}{x^2 + 1}$$ 9. Example: Find the derivative of $g(x) = 3^x$. 10. Using the formula for exponential functions: $$g'(x) = 3^x \ln(3)$$ This completes the explanation and examples for derivatives of logarithmic and exponential functions.