Derivatives of logarithmic functions rules and theory related to it
In order to obtain the derivatives of logarithmic functions, we have to start with the following theorem about functions that are inverse of each other. If f(x) and g(x) are inverses to each other, then we have derivatives of log functions g'(x) = 1 / f'(g(x)). This fact is very important and useful in finding the derivative of log function. We know that the natural exponential function and the log function are inverses of each other and we also know the derivative of the first one i.e. of ax as ax ln a. Now considering f(x) = ex and g(x) = ln x, lets use the previously discussed property, g'(x) = 1 / f'(g(x)) = 1 / eg(x) = 1 / eln x = 1/x. Thus derivative of logarithmic functions ln x is 1/x for x > 0.
It is very important to be aware of the derivatives of logarithmic functions rules. Observe the condition x > 0. This condition is required by the logarithm and must also be required for its derivative. Next, we will find the derivative of the general log function. In order to reach this formula, we need to use the change of base formula. The formula is loga x = ln x / ln a. Having this relations under considerations, it will be very simple to reach the differentiation formula. Derivative of logarithmic function loga x = (1 / ln a) (derivative of ln x) = 1 / xln a. An important property that we used in the previous relation was the fact that both ln(a) and a were constants, and ln(a) could be brought out of the derivative. Considering all the formulas deducted before, we reach to the final formula stated above.
Short recapitulation of derivatives of logarithmic functions formulas
In order to make sure you have learned something from this article, we will make a short selection of the derivatives of logarithmic functions formulas we have previously demonstrated.
Some of the common derivatives of logarithmic functions examples
We have deducted and demonstrated the upper formulas, now it's time to use them in derivatives of logarithmic functions examples as given below.
(1) Find the derivative of a logarithmic function loga x2:
We have derivative of loga x = 1 / x ln a. Here we also need to take the derivative of x2 that comes to be 2x. So final answer is 2x / x ln a = 2 / ln a.
(2) Find the derivative of logx x:
We know that log of anything with the same base is equal to 1 and derivative of constant 1 is zero. Hence the answer is zero.
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