Hi there. In this math post, I cover calculus derivatives of exponential functions. It is assumed that the reader is comfortable with logarithms, exponents, pre-calculus algebra and basic calculus derivatives.
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An exponential function is a function that has a number base and the independent variable is the exponent that corresponds to the number base. One example of an exponential function is 2 to the power of x. The variable x is the input variable for f(x).
I also show a table of values for two to the power of x. There is a Desmos screenshot for the graph of 2 to the x.
| x | y = 2^x |
|---|---|
| -2 | 1/4 = 0.25 |
| -1 | 1/2 = 0.5 |
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
Given a number a that is non-zero, the general form of the exponential function is:
In the math topic of calculus, the derivative is the slope of a line at a point of the function. This section focuses on obtaining the derivative of exponential functions algebraically.
Given the general form of the exponential function, the corresponding derivative is given as follows.
Note that ln a is the natural logarithm of the number a.
Example One
What is the derivative of 10 to the power of x?
Example Two
At x = 1, what is the derivative of 5 to the power of x? In addition, what is the equation of the tangent line that passes through x = 1?
Use this value of m = 5 ln5 or 8.0472 as the slope for the equation of this line. The equation of a line is of the form y = mx+ b.
A point must be selected in order to find the value of the y-intercept b for the line. Use f(1) = 5 to obtain the point of (1, 5). The point (1, 5) corresponds with x = 1 and y = 5 to help solve for b.
The equation of the line that passes through x = 1 for 5 to the power of x is y = 5 ln 5x - 3.0472.
In the two examples above, the exponent was just x. What if this exponent in the exponential function is something more complicated than just x? What if the exponent was something like 2x - 1, x^2 + 5?
The form of the exponential function with the exponent being a function of x is.
The derivative of the function in this case does use the chain rule. It is based on the exponent g(x).
Example One
Determine the derivative of 7 to the power of 2x + 1.
Example Two
Determine the derivative of 10 to the power of x^3 + 2x.
Thank you for reading.
