## Logarithmic Differentiation

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# Logarithmic Differentiation

The method of logarithmic differentiation is used to differentiate functions of the form y = f (x)

log y = 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] = x

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^{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 getlog y = log (x

^{x}) = x log xDifferentiating 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] = x

^{x }(1 + logx).For more help in

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