Theorem
Given that y = ln x, for x > 0, prove from first principles (or using the derivative of the exponential function) that:
dy/dx = 1/x
Answer:
Proof
Given the equation:
y=lnx
By the definition of a logarithm, we can express x in terms of y as:
x=e^y
Differentiating both sides with respect to y:
dx/dy​=e^y
Since ey=x from our initial step, we can substitute x back into the expression:
dx/dy=x
Using the rule for the derivative of an inverse function, dxdy​=dx/dy1​:
dy/dx​=1/x​
Hence, the derivative of lnx is 1/x​.
■ Q.E.D