Definition: An exponential function is of the type: y = a^x, where a is called the base of the function and a belongs to positive real numbers. Here, x is called the exponent of the base and x belongs to all real numbers. When the base of the exponential is the natural base e, then the exponential function is called the natural exponential function. It is y = e^x. Here the range of the function is positive real numbers. Since what ever the value of x, y cannot be negative or zero. The graph of this function would be like this:
From the above graph we see that there is an asymptote at y=0, that is because whatever the value of x, y can never be = 0.
Derivatives of exponential functions:
By definition of derivatives we know that for a function f(x), the derivative of f(x) denoted by f’(x) is given by the formula:
f’(x) = lim(h->0) [(f(x+h) – f(x))/h]
Using this definition we can find the derivative of the above function. So the derivative of e^x can be calculated as follows:
f(x) = e^x, f(x+h) = e^(x+h)
= > (d/dx) e^x = lim(h->0) [(f(x+h) – f(x))/h]
= > = lim(h->0) [(e^(x+h) - e^x)/h]
= > = lim(h->0) [(e^x * e^h – e^x)/h]
= > = e^x lim(h->0) [(e^h – 1)/h]
= > = e^x
Therefore we see that derivative of e^x is the same e^x.
The derivative of f(x) = e^x represents the slope of tangent of the curve at any point x=c. Therefore if we were to find the slope of tangent at x=c then that would be (d/dx) e^x |x=c, which is e^x|x=c, so slope = e^c.
Let us look at some other examples of derivatives of this natural exponential functions:
EX: Find the derivative of the function f(x) = e^2x
Solution: Here we need to use the chain rule.
Let g(x) = 2x.
Then f(g) = e^g
So f’(g) = e^g
And g’(x) = 2
Therefore, f’(x) = f’(g) * g’(x)
= e^g * 2
= 2e^g
But we had assumed g = 2x
So, f’(x) = 2e^(2x)
Thus the derivative of e^2x = 2e^2x.
On similar lines we can say that, derivative of e^ax = ae^x, where a is a constant.
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