A derivative of exponential functions

You already know the definition of a derivative of a function and you know how to find a derivative of polynomials, trigonometric, logarithmic functions. In this topic, we will consider another type of functions: exponential functions, and learn how to find the derivatives of such functions.

An exponential function

An exponential function is a function

$$f(x) = a^x$$where $a$ is called the base of the degree, and $x$ is the exponent. Notice that $a \geqslant 0$.

A derivative of this function is$$(a^x)' = a^x \cdot \ln a$$So, we take the function $a^x$ and multiply it by the natural logarithm of $a$.

Let's practice!

Consider a function $f(x) = 8^x$. We use the rule that we discussed above to find the derivative of this function:

$$(8^x)' = 8^x \cdot \ln 8$$Let's consider another example: $g(x) = 3^{x^2}$. We find the derivative of $g(x)$ using the formula for the exponential function, and since the degree is a more complex expression than just $x$, we also multiply it by the derivative of the degree, that is

$$\left(3^{x^2}\right)' = 3^{x^2} \cdot \ln 3 \cdot (x^2)' = 3^{x^2} \cdot 2x \cdot \ln 3$$Now let's look at another example: $h(x) = \tan(3^x)$. It is a composite function, so we need the chain rule to find the derivative:

$$\left(\tan(3^x)\right)' = \dfrac{1}{\cos^2 3^x} \cdot (3^x)' = \dfrac{3^x \ln 3}{\cos^2 3^x}$$

Conclusion

In this topic we have learned about an exponential function and its derivative. If we have a function $f(x) = a^x$ and want to find its derivative, we take the function $a^x$ and multiply it by the natural logarithm of $a$. Also, we have considered a few examples.