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Chapter 6 | Applications of Integration
Figure 6.75 (a) When x >1, the natural logarithm is the area under the curve y =1/ t from 1 to x . (b) When x <1, the natural logarithm is the negative of the area under the curve from x to 1.
Notice that ln1=0. Furthermore, the function y =1/ t >0 for x >0. Therefore, by the properties of integrals, it is clear that ln x is increasing for x >0. Properties of the Natural Logarithm Because of the way we defined the natural logarithm, the following differentiation formula falls out immediately as a result of to the Fundamental Theorem of Calculus.
Theorem 6.15: Derivative of the Natural Logarithm For x >0, the derivative of the natural logarithm is given by d dx ln x = 1 x .
Theorem 6.16: Corollary to the Derivative of the Natural Logarithm The function ln x is differentiable; therefore, it is continuous.
A graph of ln x is shown in Figure 6.76 . Notice that it is continuous throughout its domain of (0, ∞).
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