for f(x)=2^x is f(x) an increasing function.?

To determine if the function f(x) = 2^x is increasing, we need to check if the function values increase as the input values increase.

One way to do this is by taking the derivative of the function and determining its sign. If the derivative is non-negative, then the function is increasing; if the derivative is non-positive, then the function is decreasing.

To find the derivative of f(x) = 2^x, we can use the exponential function derivative rule, which states that the derivative of a^x with respect to x is a^x * ln(a).

So, applying this rule to our function f(x) = 2^x, we have:

f'(x) = 2^x * ln(2)

Since ln(2) is a positive constant, we can observe that f'(x) = 2^x * ln(2) is always positive. This means that the derivative is always greater than zero, so f(x) = 2^x is an increasing function.

Therefore, we can conclude that f(x) = 2^x is an increasing function.