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.

flashcards

QuestionAnswer
What is the gradient of an exponential graph proportional to at any point?It is directly proportional to the value of the function at that point (or the y coordinate).
Written as: \frac{dy}{dx} = ky
What special property does the graph of y=e^x have regarding its gradient?Its gradient at any point is identical to the y coordinate of that point.
Written as: \frac{dy}{dx} e^x = e^x
For which base does the property “gradient equals y-value” only work?It only works for base e.
What is the gradient rule for the function y=e^{kx}?The gradient is proportional to both the value of the function and the coefficient k.
Written as: \frac{dy}{dx} e^{kx} = k e^{kx}
For the function y=e^{3x}, what is the gradient at any point?The gradient is 3 times the y value at that point.