How would you like to proceed? The derivative of an exponential function is a constant times itself. Unfortunately it is beyond the scope of this text to compute the limit However, we can look at some examples. Consider and :. As we can already see, some of these limits will be less than and some larger than.
Somewhere between and the limit will be exactly. This happens when We will define the number by this property in the next definition:. Hence is its own derivative. In other words, the slope of the plot of is the same as its height, or the same as its second coordinate. Hence we have derived the derivative of exponential function using the first principle of derivatives.
We know that the derivative of e x is e x. Since the first derivative of exponential function e x is e x , therefore if we differentiate it further, the derivative will always be e x. Hence the nth derivative of e x is e x.
Yes, the derivative of exponential function e x is the exponential function e x itself. Learn Practice Download. What is Derivative of Exponential Function? Derivative of Exponential Function Formula 3. In the system of natural logarithms, in which e is the base, we have the simplest constant possible, namely 1. Example 1. Problem 1. Calculate the derivative of e x 2.
To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" "Reload". Do the problem yourself first! Problem 2. Calculate the derivative of the following. According to the product rule. Example 4. Find the derivative of ln x 2. According to the 2nd Law :. Problem 3. Differentiate the following. Problem 5. The derivative of log a x. That derivative approaches 0, that is, becomes smaller. That derivative becomes larger. If n is irrational, a rational approximation will be necessary.
Problem 7.
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