- #Derivative of log and exponential functions how to#
- #Derivative of log and exponential functions plus#
f prime of x equals lna and so we summarize this by saying the derivative of an exponential function, the derivative with respect to x of a to the x equals natural log of a times a to the x. Now this actually is going to be constant with respect to h, as h goes to zero nothing happens to a to the x but this thing approaches ln of a and so thatÂ’s our derivative. Now one of the things that we showed in a previous episode was that the limit of this quantity was actually the natural log of a. So this will be the limit as h approaches zero of a to the h minus 1 over h times a to the x.
![derivative of log and exponential functions derivative of log and exponential functions](https://i.pinimg.com/736x/3b/ef/55/3bef554cc832fe513172a13d51450301--logarithmic-functions-calculus.jpg)
Let me pop that up into the limit because we'll call it the derivative of f is the limit as h approached zero of the difference quotient. So I can write this as and I'll pull it out to the right and what's left? In this term a to the h minus and then 1 all over h, so this is my simplified difference quotient.
![derivative of log and exponential functions derivative of log and exponential functions](https://images.slideplayer.com/16/5230896/slides/slide_3.jpg)
#Derivative of log and exponential functions plus#
Now a to the x plus h is the same as a to the x times a to the h by a property of exponents, minus a to the x, and now you can see that a to the x is a common factor in both terms of the numerator and it could be factored out. So we just have to simplify this just a little bit. Now for this function f of x plus h is a to the x plus h and f of x is a to the x. Let's find this derivative f prime of x now you'll recall that usually the first thing we do is we work on the difference quotient f of x plus h minus f of x over h. Now recall that exponential function is one of the form f of x equals a to the x where a is a positive constant but not 1. From this definition, we derive differentiation formulas, define the number e, and expand these concepts to logarithms and exponential functions of any base.We've been talking about derivatives of different kinds of functions well one class of functions that's really important is exponential functions. This definition forms the foundation for the section. We begin the section by defining the natural logarithm in terms of an integral.
![derivative of log and exponential functions derivative of log and exponential functions](https://slidetodoc.com/presentation_image/47a593446a8e0b47db77df5abf42e55d/image-25.jpg)
By the end of the section, we will have studied these concepts in a mathematically rigorous way (and we will see they are consistent with the concepts we learned earlier). We now have the tools to deal with these concepts in a more mathematically rigorous way, and we do so in this section.įor purposes of this section, assume we have not yet defined the natural logarithm, the number e, or any of the integration and differentiation formulas associated with these functions. The definition of the number e is another area where the previous development was somewhat incomplete.
#Derivative of log and exponential functions how to#
For example, we did not study how to treat exponential functions with exponents that are irrational. However, we glossed over some key details in the previous discussions. We already examined exponential functions and logarithms in earlier chapters.
![derivative of log and exponential functions derivative of log and exponential functions](https://image.slidesharecdn.com/lesson08-derivativesoflogarithmsandexponentialfunctionsslides-100601221744-phpapp01/95/lesson-14-derivatives-of-logarithmic-and-exponential-functions-73-728.jpg)