![derivative of log 5 derivative of log 5](https://image.slidesharecdn.com/lesson14-derivativesoflogarithmicandexponentialfunctions011slides-110322222511-phpapp01/95/lesson-14-derivatives-of-logarithmic-and-exponential-functions-slides-35-728.jpg)
![derivative of log 5 derivative of log 5](https://s3mn.mnimgs.com/img/shared/discuss_editlive/1915302/2012_02_13_13_07_01/mathmlequation3844296310700307069.png)
Is therefore a pullback of the invariant form. Before taking the derivative, we will expand this expression. Is invariant under dilation (replacing X by aX for a constant). This also includes the natural logarithmic function y ln. ( log u v ) ′ = ( log u + log v ) ′ = ( log u ) ′ + ( log v ) ′. Implicit differentiation, which we explored in the last section, can also be employed to find the derivatives of logarithmic functions, which are of the form y log a. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does not take values in the positive reals.
![derivative of log 5 derivative of log 5](https://i.ytimg.com/vi/HORYwxL0c34/hqdefault.jpg)
2 Computing ordinary derivatives using logarithmic derivatives.