Derivative of log(x) - Proof and Explanation

Proof

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

1. Rewrite $y=\ln (x)$ as $x=e^y$.

2. Differentiate both sides with respect to $x$:

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

3. This gives:

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

4. Solving for $\frac{dy}{dx}$:

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

5. Substitute $e^y=x$:

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

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

Explanation

The natural logarithm function, $\ln (x)$, is the inverse of the exponential function $e^x$. To find the derivative, we start by setting $y=\ln (x)$. 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 $\ln (x)$ is $\boxed{\frac{1}{x}}$.

Q.E.D.