Exponential Derivatives

As a brief referesher, Exponential Functions are commonly expressed as \(f(x) = a^x\), \(a > 0\). They are commonly used to model problems related to Growth and Decay. One of the most common forms of the exponential function is the Natural Exponential Function, \(f(x) = e^x\), where \(e = 2.71828\)..., also referred to as Euler's Constant.

The derivative of an Exponential Function can be expressed as such:

\(\cfrac{d}{dx}\cdot a^x = \textcolor{red}{a^x}\cdot \textcolor{blue}{\text{ln}(a)}\)

Likewise, the derivative of a Natural Exponential Function can be expressed as such:

\(\cfrac{d}{dx}\cdot e^x = e^x\)

As you can see, the derivative of this function is the exact same as the original function. This is because it also takes the product of the Natural Logarithm, \(\text{ln}(e)\). However, since \(\text{ln}(e) = 1\), it simplifies itself to the form of its original function.


Example

Find the derivative of the following exponentials.

  1. \(y = 5^x\)
  2. \(y = 4e^x\)

i. We can use the Exponential Formula to determine the derivative:

\(\cfrac{d}{dx}\cdot a^x = a^x\cdot \text{ln}(a)\)

In this equation, we can determine that \(a = 5\). As a result, we can express the derivative as:

\(y' = 5^x\cdot \text{ln}(5)\)

Therefore, we can determine that the derivative of the original equation is \(y' = 5^x\cdot \text{ln}5\).


ii. We can use the Natural Exponential Formula to determine the derivative:

\(\cfrac{d}{dx}\cdot e^x = e^x\)

Since the derivative of a Natural Exponential is the same as the original equation, it can be written as:

\(y' = 4e^x\)

Therefore, we can determine that the derivative of the original equation is \(y' = 4e^x\).


Find the derivative of the following functions.

\(y = x^5 \cdot 5^x\)


\(y = \cfrac{2^{4x}}{x^3}\)


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