Are exponential functions differentiable?

Both approaches have serious pedagogical faults, which are discussed later in this paper. Our proof that exponential functions are differentiable provides the missing link that legitimizes the “early transcendentals” presentation. ax is positive and continuous, ax is increasing if a > 1, ax is decreasing if a < 1.

What are the concepts of exponential function?

Key Concepts An exponential function is defined as a function with a positive constant other than 1 raised to a variable exponent. A function is evaluated by solving at a specific input value. An exponential model can be found when the growth rate and initial value are known.

How to differentiate a derivative of an exponential function?

Derivatives of Exponential Functions In order to differentiate the exponential function f (x) = a^x, f (x) = ax, we cannot use power rule as we require the exponent to be a fixed number and the base to be a variable.

Can a power rule differentiate an exponential function?

In order to differentiate the exponential function we cannot use power rule as we require the exponent to be a fixed number and the base to be a variable. Instead, we’re going to have to start with the definition of the derivative:

Which is the best definition of an exponential function?

We use the following definition of the exponential function: Let’s define A:R∗ → R A(h) = exp(h)− 1 h −1 We’re going to show that limh→0A(h) = 0. This will imply that limh→0exp (h)−1 h = 1 and consequently, that exp′(0) = 1. Let h ∈[− 1,1]∖{0}. We define the sequence (uk)k∈N∗ by uk = (1+ h k)k− 1 h − 1

Why do we need to know the derivative of f ( x )?

Now let’s notice that the limit we’ve got above is exactly the definition of the derivative of f (x) = ax f ( x) = a x at x =0 x = 0, i.e. f ′(0) f ′ ( 0). Therefore, the derivative becomes, So, we are kind of stuck. We need to know the derivative in order to get the derivative! There is one value of a a that we can deal with at this point.