Exponential graph gradient

An important property of an exponential graph is that the *gradient of it is directly proportional to the value of y at that point.

That means that, for a function in the form f(x)=a^x, we can write the gradient as f'(x)=ka^x (where k is a constant of proportionality).

In other words, the gradient of the graph is the y coordinate multiplied by something.

Gradient of e^x

There’s a special case of this when the base is e. The gradient at any point is exactly the same as the y value at that point.

We’ll learn more about that in the base e exponentials topic, but for now, just remember that the gradient of e^x is e^x itself.

Question	Answer
What is the relationship between the gradient of an exponential graph and its y-value?	The gradient is directly proportional to the value of y at that point.
For a function f(x)=a^x, how can the gradient f'(x) be written?	f'(x)=ka^x, where k is a constant of proportionality.
What is special about the gradient of the graph of e^x?	The gradient at any point is exactly the same as the y value at that point, so the gradient of e^x is e^x itself.