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.