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Logarithmic Differentiation


Consider now a technique called logarithmic differentiation, which is useful for differentiating compound functions of products, quotients and powers.

Example

The derivative of

It is relatively difficult to calculate directly. However, if we first take the natural logarithm on both sides and then use its properties, we can write:

Differentiating both sides from x, results

So solving for dy / dx and using we get

NOTE.Once 1n y is set only to y> 0, the logarithmic differentiation of y = f(x) is valid only at intervals where f(x) is positive. Thus, the derivative shown in the example is valid in the range (2, + ), since the given function is positive for x> 2. However, the formula is really valid also in the range (- , 2). This can be seen by taking absolute values ​​before proceeding with logarithmic differentiation and noting that is set to all y except in y = 0. If we do this and simplify using the logarithm and absolute value properties, we get

Differentiating both sides from x gives rise to, and therefore results in. In general if the derivative of y = f(x) is obtained by logarithmic differentiation, then the same formula for dy / dx result in taking or not absolute values ​​first. Thus, a derivative formula obtained by logarithmic differentiation will be valid, except at the points where f(x) is zero. The formula may also be valid at those points, but it is not guaranteed.

Next: Derivatives of Exponential Functions