Solve $\frac{d}{dx}\ln x$

Step-by-step solution

1. Step 1 · Definition of the natural log
   $$y=\ln x⟺e^y=x$$

   Rewrite so implicit differentiation can be used.

2. Step 2 · Implicit differentiation
   $$e^y\frac{dy}{dx}=1$$

   Differentiate both sides with respect to x.

3. Step 3 · Substitute back
   $$\frac{dy}{dx}=\frac{1}{e^y}=\frac{1}{x}$$

   Since eʸ = x, the derivative is 1/x for x > 0.