## Proof

Let $y=\mathrm{ln}(x)$. To find the derivative, we use the definition of the natural logarithm:

Rewrite $y=\mathrm{ln}(x)$ as $x={e}^{y}$.

Differentiate both sides with respect to $x$:

$$\frac{\text{d}}{\text{d}x}(x)=\frac{\text{d}}{\text{d}x}({e}^{y})$$This gives:

$$1={e}^{y}\frac{dy}{dx}$$Solving for $\frac{dy}{dx}$:

$$\frac{dy}{dx}=\frac{1}{{e}^{y}}$$Substitute ${e}^{y}=x$:

$$\frac{dy}{dx}=\frac{1}{x}$$

Thus, the derivative of $\mathrm{ln}(x)$ is:

$$\frac{\text{d}}{\text{d}x}\mathrm{ln}(x)=\overline{)\frac{1}{x}}$$## Explanation

The natural logarithm function, $\mathrm{ln}(x)$, is the inverse of the exponential function ${e}^{x}$. To find the derivative, we start by setting $y=\mathrm{ln}(x)$. This means that $x$ can be expressed as ${e}^{y}$.

When differentiating both sides with respect to $x$, the left side simply becomes $1$. For the right side, using the chain rule, the derivative of ${e}^{y}$ with respect to $x$ is ${e}^{y}\frac{dy}{dx}$.

Setting these equal gives the equation $1={e}^{y}\frac{dy}{dx}$. We then solve for $\frac{dy}{dx}$, which involves dividing both sides by ${e}^{y}$. This simplifies to $\frac{1}{{e}^{y}}$.

Finally, since ${e}^{y}=x$ (from our original substitution), we can substitute back to get $\frac{dy}{dx}=\frac{1}{x}$.

Therefore, the derivative of $\mathrm{ln}(x)$ is $\overline{)\frac{1}{x}}$.

*Q.E.D.*