Louisiana State University
Section 3.9 Derivatives of Logarithmic and Exponential FunctionsTopic 1: Review of Logarithmic and Exponential FunctionsInverse Properties for ex and ln?xeln?x=x, for x>0, and ln?ex=x, for all x.y=ln?x if and only if x=ey.For all real numbers x and b>0, bx=eln?bx=exln?b.Properties of Logarithmslogbxy=logbx+logby, for x>0 and y>0logbxy=logbx-logby, for x>0 and y>0logbxy=ylogb?x, for x>0Topic 2: Derivatives Involving Natural Logarithm FunctionsTheoremDerivative of ln?x and ln(ux)ddxln?x=1x, for x>0ddxln?|x|=1x, for x≠0If u is differentiable at x and u(x)≠0, thenddxln?|u(x)|=u'(x)u(x).Topic 3: Logarithmic DifferentiationConsider the function fx=x3-143x-1x2+4.In order to find the derivative of f, we would need to use the quotient rule, product rule, and chain rule and then simplify the result. In cases such as this, the properties of logarithms reviewed at the beginning of this section are useful for differentiating a ic 4: Derivatives Involving Natural Exponential FunctionsTheoremDerivative of ex and euxThe function fx=ex is differentiable for all real numbers x, andddxex=ex.If u is differentiable at x, thenddxeux=eux?u'(x).Topic 5: Derivatives of General Logarithmic and Exponential FunctionsTheoremDerivative of bx If b>0 and b≠1, thenddxbx=bxln?b, for all x.TheoremDerivative of logb?x If b>0 and b≠1, thenddxlogb?x=1xln?b, for x>0 and ddxlogb?|x|=1xln?b, for x≠0. ................
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