The derivative of the natural logarithm is:

ddx[ln⁡(x)]=1x,x>0.\frac{d}{dx}[\ln(x)]=\frac{1}{x},\quad x>0.dxd​[ln(x)]=x1​,x>0.

Quick Scoop

Think of ln⁡(x)\ln(x)ln(x) as a smooth curve that grows slowly as xxx increases. Its slope at any point xxx is exactly 1/x1/x1/x:

  • At x=1x=1x=1, the slope is 111.
  • At x=2x=2x=2, the slope is 1/21/21/2.
  • As xxx gets larger, the slope gets smaller, matching the idea that ln⁡(x)\ln(x)ln(x) grows more and more slowly.

If you meant a more complicated function like ln⁡(g(x))\ln(g(x))ln(g(x)), then by the chain rule,

ddx[ln⁡(g(x))]=g′(x)g(x).\frac{d}{dx}[\ln(g(x))]=\frac{g'(x)}{g(x)}.dxd​[ln(g(x))]=g(x)g′(x)​.

You can use this pattern for things like ln⁡(x2+1)\ln(x^2+1)ln(x2+1), ln⁡(3x)\ln(3x)ln(3x), or ln⁡(ln⁡x)\ln(\ln x)ln(lnx).