Exponential function gradient

The gradient of an exponential graph at any point is directly proportional to the value of the function at that point (or the y coordinate of the graph).

This can be written as:

\frac{dy}{dx} = ky

Gradient of e^x

The graph of y=e^x has an interesting property - its gradient at any point is identical to the y coordinate of that point.

For example, at x=4, the point on the graph is (4, e^4). The gradient at this point is also e^4.

This can be written as:

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

Reminder: this ONLY works for base e. Other bases will have different gradients to their y values.

Gradient of e^{kx}

When we have an exponential function with a coefficient in the exponent, i.e. y=e^{kx}, the gradient is proportional to both the value of the function and the coefficient k.

We can write this as:

\frac{dy}{dx} e^{kx} = k e^{kx}

For example, for the function y=e^{3x}, the gradient at any point is 3 times the y value at that point.