Chapter 2: Differentiation
Section 2.6: Derivatives of the Exponential and Logarithmic Functions
Example 2.6.5
Use logarithmic differentiation to obtain the derivative of Fx=ux⋅vx⋅wx.
Solution
Table 2.6.5(a) contains the solution. Note that the outcome is consistent with the comment about the pattern for the Product rule stated in Section 2.3.
Fx
=ux⋅vx⋅wx
lnFx
=lnux+lnvx+lnwx
ddx lnFx
=ddxlnux+ddxlnvx+ddxlnwx
F′xFx
=u′xux+v′xvx+w′xwx
F′x
=uxvxwxu′xux+v′xvx+w′xwx
=u′⋅v⋅w+u⋅v′⋅w+u⋅v⋅w′
Table 2.6.5(a) Logarithmic differentiation with three factors
Take the natural log of both sides. The log of the product on the right is the sum of logs. Differentiate both sides. On the left, the derivative of F is F′/F while on the right, the derivatives of the logs yield a sum of fractions of the form derivative divided by function. Multiply through by F, which is itself the original product, and simplify.
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