Base e exponentials

There’s a special case of exponential functions, where we use a base of .

You can find more about in the euler’s number topic. It’s a number with some special properties, and it’s about .

Reciprocal exponentials

If, instead of , we have , we can rewrite it as .

That’s a decay graph, as the base is less than (because is less than ).

Gradient of

The gradient of is exactly the same as the value at that point.

For example, at the point , the gradient is (because ). At the point

Gradient of

doesn’t exactly differentiate to , but it does differentiate to a constant multiple of : .

Gradient of

The gradient of is - it’s the same as the rule for , but with a negative constant instead of a positive one.

Gradient of

This is basically just saying that is a negative constant, so the gradient of is - it follows the same rule as , but with a negative constant instead of a positive one.

Gradient of

The gradient of is - it’s the same as the rule for , but instead of being a constant, it’s , which is the derivative of .