6. Derivative of the Exponential Function
by M. Bourne
The derivative of ex is quite remarkable. The expression for the derivative is the same as the expression that we started with; that is, ex!
What does this mean? It means the slope is the same as the function value (the y-value) for all points on the graph.
Example: Let's take the example when x = 2. At this point, the y-value is e2 ≈ 7.39.
Since the derivative of ex is ex, then the slope of the tangent line at x = 2 is also e2 ≈ 7.39.
We can see that it is true on the graph:
Let's now see if it is true at some other values of x.
We can see that at x = 4, the y-value is 54.6 and the slope of the tangent (in red) is also 54.6.
At x = 5, the y-value is 148.4, as is the value of the derivative and the slope of the tangent (in green).
Other Formulas for Derivatives of Exponential Functions
If u is a function of x, we can obtain the derivative of an expression in the form eu:
If we have an exponential function with some base b, we have the following derivative:
`(d(b^u))/(dx)=b^u\ ln\ b(du)/(dx)`
Find the derivative of y = 103x.
Find the derivative of y = ex2.
Find the derivative of y = sin(e3x).
Find the derivative of y = esin x.
Find the derivative of
1. Find the derivative of y = 10x2.
2. Find the derivative of
3. Find the derivative of
4. Show that
satisfies the equation