The method of finding the derivatives of some difficult functions using logarithms is called Derivatives of Logarithmic Functions. Differentiating the logarithm of a given function can be easier in some circumstances than differentiating the function itself. The derivatives become simple with the right use of logarithm characteristics and chain rule discovery. This principle is applicable to nearly all non-zero, differentiable functions in nature.
We’ve learned how to differentiate a wide range of functions so far, including trigonometric, inverse, and implicit functions. We’ll look at derivatives of logarithmic function using the first principles of derivatives in this section. Logarithmic functions can be used to rescale huge numbers and are especially useful when rewriting complex equations.
Therefore, in calculus, the differentiation of some complicated functions is accomplished by obtaining logarithms and then using the logarithmic derivative to solve such a function.
On that note, let’s learn about the derivative of the logarithmic function using the first principle and other related concepts in depth.
Logarithmic Differentiation Formula
The chain rule is used to calculate the derivatives of logarithmic functions. However, we may use a logarithmic function to generalize it to any differentiable function. We can only differentiate log under the base e,e, but we can differentiate under other bases as well.
y = f(x) = [u(x)] are equations that have the form y = f(x) = [u(x)].
The principle of logarithmic differentiation can be used to solve v(x). A function’s log differentiation formula is as follows:
xx(1+ln x) = d/dx(xx)
Here’s where you may get a complete list of differentiation formulas.
We take on both sides of the given equation when differentiating functions of this type.
Therefore, taking log on both sides we get,log y = log[u(x)]{v(x)}
log y = v(x)log u(x)
The only stipulation for utilizing logarithmic differentiation methods is that f(x) and u(x) must be positive, as logarithmic functions can only be defined for positive numbers.
The essential features of real logarithms apply to logarithmic derivatives in general.
Derivatives of Logs
The derivatives of logs will be discussed. The derivatives of both common and natural logarithms, in other words. We already know that the log ax derivative is 1 / (x ln a). A common logarithm is defined as a log ax. The natural logarithm, on the other hand, is a different form of a logarithm. ln x is the symbol for it. It is a base “e” logarithm, therefore it can be expressed as ln x = log ex. We now have
1 / d/dx (loga x) (x ln a)
On both sides, substitute a = e. Then there’s:
1 / d/dx (loge x) (x ln e)
ln e = 1 according to natural logarithm properties.
d/dx (loge x) = 1 / (x 1) d/dx (loge x) = 1 / (x 1) d/dx (loge x)
As a result, d/dx (loge x) = 1.
Replacing loge x with ln x back, we get d/dx (ln x) = 1/x.
Method to Solve Logarithmic Functions
To find the differentiation of logarithmic functions, follow the techniques outlined here.
- First, find the natural log of the function to be differentiated.
- Distribute the terms that were originally grouped together in the original function and were difficult to distinguish using features of logarithmic functions.
- Now, take the obtained equation and divide it by two.
- Finally, to obtain the desired derivative, multiply the given equation by the function itself.
Key Concepts
- This function is differentiable everywhere and has a formula for its derivative, based on the premise that the exponential function y=bx,b>0 is continuous everywhere and differentiable at 0.
- The derivative of y=lnx can be found using a formula, and the relationship log b x=ln x ln b allows us to extend our differentiation formulas to include logarithms with any bases.
- By taking the natural logarithms of both sides and leveraging the features of logarithms before differentiating, we can differentiate functions of the form y=g(x)f(x) or highly complex functions.
