Base e exponentials

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

You can find more about e in the euler’s number topic. It’s a number with some special properties, and it’s about 2.71828.

Reciprocal exponentials

If, instead of y=e^x, we have y=e^{-x}, we can rewrite it as y=\frac{1}{e^x}.

That’s a decay graph, as the base is less than 1 (because \frac{1}{e} is less than 1).

Gradient of e^x

The gradient of e^x is exactly the same as the y value at that point.

\frac{d}{dx}e^x = e^x

For example, at the point (0, 1), the gradient is 1 (because e^0=1). At the point

Gradient of e^{kx}

e^{kx} doesn’t exactly differentiate to e^{kx}, but it does differentiate to a constant multiple of e^{kx}: ke^{kx}.

\frac{d}{dx}e^{kx} = ke^{kx}

Gradient of e^{-x}

The gradient of e^{-x} is -e^{-x} - it’s the same as the rule for e^{kx}, but with a negative constant instead of a positive one.

\frac{d}{dx}e^{-x} = -e^{-x}

Gradient of e^{-kx}

This is basically just saying that k is a negative constant, so the gradient of e^{-kx} is -ke^{-kx} - it follows the same rule as e^{kx}, but with a negative constant instead of a positive one.

\frac{d}{dx}e^{-kx} = -ke^{-kx}

Gradient of e^{f(x)}

The gradient of e^{f(x)} is f'(x)e^{f(x)} - it’s the same as the rule for e^{kx}, but instead of k being a constant, it’s f'(x), which is the derivative of f(x).

\frac{d}{dx}e^{f(x)} = f'(x)e^{f(x)}

flashcards