The derivative of an exponential function is equal to the derivative of the exponent, multiplied by the original function and by the natural logarithm of the base.
That is, in mathematical terms, we would have the following formula:
In the function above, z is the base and y is a function of x, whose derivative can be calculated as explained in our article on the derivative of a function.
We must remember that a derivative is a mathematical function that allows us to calculate the rate of change of a (dependent) variable. This, when a variation is registered in another variable (which would be the independent one) that affects it.
Cases of the exponential function
The exponential function presents two particular cases:
- When the exponent is x, the derivative of this is 1. Therefore, the derivative of the exponential function is equal to this same function times the natural logarithm of the base, as we see below:
- When the base is the constant e, its natural logarithm is 1. Therefore, the derivative of the exponential function would be equal to the derivative of the exponent times the original function.
Examples of derivative of an exponential function
Let’s look at some worked-out exponential function examples:
Now, a second example a little more complex:
Now, let’s look at an example where the exponent is a trigonometric function: