Before getting to the derivative that exponential function, let united state recall the ide of one exponential function which is given by, f(x) = ax, a > 0. Among the well-known forms of the exponential function is f(x) = ex, whereby 'e' is "Euler's number" and also e = 2.718.... The derivative that exponential duty f(x) = ax, a > 0 is given by f'(x) = ax ln a and also the derivative the the exponential duty f(x) = ex is provided by f'(x) = ex.

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In this article, us will research the ide of the derivative the the exponential role and that formula, proof, and also graph together with some solved instances to recognize better.

1.What is Derivative that Exponential Function?
2.Derivative of Exponential role Formula
3.Derivative that Exponential duty Proof
4.Graph that Derivative of Exponential Function
5.FAQs top top Derivative of Exponential Function

What is Derivative the Exponential Function?


The derivative the exponential duty f(x) = ax, a > 0 is the product the exponential function ax and natural log in of a, the is, f'(x) = ax ln a. Mathematically, the derivative of exponential duty is created as d(ax)/dx = (ax)' = ax ln a. The derivative the exponential role can be acquired using the first principle the differentiation utilizing the formulas of limits. The graph the derivative of exponential duty changes direction when a > 1 and when a x

Now the we know the derivative the the exponential role is offered by f'(x) = ax ln a, the derivative the exponential function ex using the same formula is offered by ex ln e = ex (because ln e = 1). Hence the derivative of exponential duty ex is the function itself, the is, if f(x) = ex, climate f'(x) = ex.


Derivative that Exponential function Formula


The formula for derivative of exponential duty is offered by,

f(x) = ax, f'(x) = ax ln a or d(ax)/dx = ax ln af(x) = ex, f'(x) = ex or d(ex)/dx = ex

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Derivative the Exponential duty Proof


Now, we will certainly prove that the derivative of exponential duty ax is ax ln a making use of the very first principle the differentiation, the is, the definition of limits. To derive the derivative that exponential function, we will certainly some formulas together as:

(f'(x)=lim_h ightarrow 0dfracf(x+h)-f(x)h)(lim_h ightarrow 0dfraca^h-1h = ln a)am × an = am+n

Using the over formulas, we have

(eginalign fracmathrmd (a^x)mathrmd x&=lim_h ightarrow 0dfraca^x+h-a^xh\&=lim_h ightarrow 0dfraca^x imes a^h-a^xh\&=lim_h ightarrow 0dfraca^x (a^h-1)h\&=a^xlim_h ightarrow 0dfrac a^h-1h\&=a^x ln aendalign)

Hence us have obtained the derivative of exponential role using the very first principle that derivatives.


Graph of Derivative that Exponential Function


The graph of exponential function f(x) = bx is enhancing when b > 1 vice versa, f(x) = bx is decreasing once b x

increases when b > 1decreases as soon as 0

Since the graph of derivative of exponential function is the slope role of the tangent to the graph the the exponential function, therefore the graph that derivative that exponential function f(x) = bx is raising for b > 0. Given listed below is the graph the the exponential role and the graph that the derivative of exponential function for b > 1 and also 1 Exponential function f(x) = ax is not defined for a d(ax)/dx = ax ln ad(ex)/dx = ex

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Derivative the Exponential duty Examples


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FAQs top top Derivative the Exponential Function


What is the Derivative of Exponential Function?

The derivative that exponential function f(x) = ax, a > 0 is the product that exponential function ax and also natural log in of a, that is, f'(x) = ax ln a.

What is the Derivative that Exponential duty ex?

The derivative the exponential duty ex is the duty itself, the is, if f(x) = ex, then f'(x) = ex.

Why is the Derivative that Exponential role ex itself?

The derivative that the exponential role ax is given by f'(x) = ax ln a. We understand that ln e = 1 and if a = e, the derivative that exponential duty ex is given by ex ln e = ex

How to find the nth Derivative that Exponential duty ex?

Since the first derivative the exponential role ex is ex, thus if we differentiate it further, the derivative will always be ex. Therefore the nth derivative the ex is ex.

Is the Derivative of Exponential duty ex the Exponential Function?

Yes, the derivative that exponential duty ex is the exponential function ex itself.

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What is the general Derivative that Exponential Function?

The general derivative the exponential role ax is provided by ax ln a.