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Logarithmic Differentiation

The method of logarithmic differentiation is used to differentiate functions of the form y = f (x)g (x) . With this method, we first take the natural logarithm of both sides of y = f (x)g (x) to obtain log y = log [ f (x)g (x) ] . After simplifying log [ f (x)g (x) ] by using properties of logarithms, we differentiate both sides with respect to x and then solve for dy/dx. This method can also be used to differentiate functions which are product of several functions. The following examples illustrate the method of logarithmic differentiation.

Example . If y = xx, find dy/dx .

Solution . The given function has the form f (x)g (x) . Taking the natural logarithm of both sides, we get

log y = log (xx) = x log x

Differentiating both sides with respect to x gives

1/y dy/dx = x (1/x) + (log x) (1) = 1 + log x

Multiplying both sides by y and then substituting xx for y, we obtain

dy/dx = y [ 1 + logx] = xx (1 + logx).

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