Before obtaining to the derivative of exponential function, let us recall the concept of an exponential role which is given by, f(x) = ax, a > 0. One of the well-known forms the the exponential role is f(x) = ex, where 'e' is "Euler's number" and e = 2.718.... The derivative that exponential role f(x) = ax, a > 0 is offered by f'(x) = ax ln a and the derivative of the exponential role f(x) = ex is offered by f'(x) = ex.
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In this article, we will study the concept of the derivative that the exponential duty and its formula, proof, and graph along with some solved instances to know better.
|1.||What is Derivative the Exponential Function?|
|2.||Derivative the Exponential duty Formula|
|3.||Derivative that Exponential duty Proof|
|4.||Graph of Derivative of Exponential Function|
|5.||FAQs on Derivative the Exponential Function|
What is Derivative that Exponential Function?
The derivative the exponential duty f(x) = ax, a > 0 is the product of exponential role ax and also natural log of a, the is, f'(x) = ax ln a. Mathematically, the derivative of exponential function is written as d(ax)/dx = (ax)' = ax ln a. The derivative of exponential duty can be obtained using the very first principle of differentiation making use of the recipe of limits. The graph that derivative the exponential role changes direction when a > 1 and also when a x
Now the we know the derivative the the exponential duty is given by f'(x) = ax ln a, the derivative that exponential duty ex making use of the same formula is given by ex ln e = ex (because ln e = 1). Thus the derivative the exponential duty ex is the duty itself, that is, if f(x) = ex, climate f'(x) = ex.
Derivative the Exponential duty Formula
The formula for derivative of exponential role is given 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
Derivative of Exponential role Proof
Now, we will prove the the derivative of exponential role ax is ax ln a using the an initial principle of differentiation, that is, the meaning of limits. To have the derivative the exponential function, we will some formulas such 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 above 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 we have derived the derivative that exponential role using the first principle of derivatives.
Graph of Derivative the Exponential Function
The graph the exponential duty f(x) = bx is raising when b > 1 whereas f(x) = bx is decreasing as soon as b xincreases as soon as b > 1decreases once 0
Since the graph of derivative the exponential duty is the slope role of the tangent come the graph of the exponential function, because of this the graph of derivative that exponential role f(x) = bx is enhancing for b > 0. Given listed below is the graph that the exponential duty and the graph of the derivative that exponential duty for b > 1 and also 1 Exponential role f(x) = ax is not identified for a d(ax)/dx = ax ln ad(ex)/dx = ex
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Derivative of Exponential function Examples
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Derivative of Exponential function Problems
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FAQs top top Derivative the Exponential Function
What is the Derivative of Exponential Function?
The derivative the exponential function f(x) = ax, a > 0 is the product that exponential function ax and natural log of a, the is, f'(x) = ax ln a.
What is the Derivative that Exponential function ex?
The derivative that exponential function ex is the role itself, the is, if f(x) = ex, climate f'(x) = ex.
Why is the Derivative that Exponential function ex itself?
The derivative that the exponential role ax is given by f'(x) = ax ln a. We recognize that ln e = 1 and also if a = e, the derivative that exponential function ex is provided by ex ln e = ex
How to discover the nth Derivative the Exponential duty ex?
Since the an initial derivative the exponential function ex is ex, thus if we differentiate it further, the derivative will always be ex. Thus the nth derivative of ex is ex.
Is the Derivative that Exponential function ex the Exponential Function?
Yes, the derivative that exponential function ex is the exponential role 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.