Why derivative of exponential is exponential?

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

What does the derivative of an exponential function represent?

Note that the exponential function f( x) = e x has the special property that its derivative is the function itself, f′( x) = e x = f( x).

What does exponential function mean in calculus?

An exponential function is one that involves a constant positive base to a variable exponent. The most basic exponential is: f(x) = ax, where a > 0 is a constant. Other variations include coefficients that scale the graph horizontally or vertically. Adding or subtracting a constant shifts the graph up or down.

Is the derivative of an exponential function a logarithmic function?

Derivatives of exponential functions involve the natural logarithm function, which itself is an important limit in Calculus, as well as the initial exponential function. The derivative is the natural logarithm of the base times the original function.

What function has a derivative that is itself?

exponential function
The derivative of an exponential function is a constant times itself. Using this definition, we see that the function has the following truly remarkable property. 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.

Why exponential function is important?

In economics exponential functions are important when looking at growth or decay. Examples are the value of an investment that increases by a constant percentage each period , sales of a company that increase at a constant percentage each period, models of economic growth or models of the spread of an epidemic.

How would you differentiate exponential function and logarithmic function?

The function E ( x ) = e x E ( x ) = e x is called the natural exponential function. Its inverse, L ( x ) = log e x = ln x L ( x ) = log e x = ln x is called the natural logarithmic function.