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What is the derivative of ln(x)?

First let y=ln(x).

Recall that the exponential function, ex, is defined as the inverse of the logarithmic function, ln(x).

To make x the subject of the formula, use the inverse function exp. This gives that x=ey.

Now, differentiate both sides with respect to y and recall that d/dx(ex)=ex. This gives dx/dy=ey.

Remember for a derivative, dy/dx=1/(dx/dy).

Therefore, dy/dx=1/ey.

Finally, from above, x=ey.

Substituting for ey we have dy/dx=1/x which is our final result.

Therefore the derivative of ln(x), is dy/dx=1/x.

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