section_path: Analytic properties/Derivative and antiderivative
section_name: Derivative and antiderivative
paragraph_text: Derivative and antiderivative
  
  Analytic properties of functions pass to their inverses. Thus, as f (x) = b is a continuous and differentiable function, so is log y. Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivative of f (x) evaluates to ln(b) b by the properties of the exponential function, the chain rule implies that the derivative of log x is given by d d x log b x = 1 x ln b. {\displaystyle {\frac {d}{dx}}\log _{b}x={\frac {1}{x\ln b}}.} That is, the slope of the tangent touching the graph of the base- b logarithm at the point (x, log (x)) equals 1/(x ln(b)).
  
  The derivative of ln(x) is 1/ x; this implies that ln(x) is the unique antiderivative of 1/ x that has the value 0 for x = 1. It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of the main reasons of the importance of the constant e.
  
  The derivative with a generalized functional argument f (x) is d d x ln f (x) = f ′ (x) f (x). {\displaystyle {\frac {d}{dx}}\ln f(x)={\frac {f'(x)}{f(x)}}.} The quotient at the right hand side is called the logarithmic derivative of f. Computing f' (x) by means of the derivative of ln(f (x)) is known as logarithmic differentiation. The antiderivative of the natural logarithm ln(x) is: ∫ ln (x) d x = x ln (x) − x + C. {\displaystyle \int \ln(x)\,dx=x\ln(x)-x+C.} Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.