Question

Find the derivative of the following functions. $$y=8^{x}$$

Short answer

Answer: The derivative of the function $$y = 8^x$$ is $$\frac{dy}{dx} = 8^x \cdot \ln(8)$$.

Step-by-step solution

Recall the rule for finding the derivative of an exponential function

For a function of the form $$y = a^x$$, where a is a constant, the derivative is given by: $$\frac{dy}{dx} = a^x \cdot \ln(a)$$ In our case, $$a = 8$$, so the rule for finding the derivative of our function is: $$\frac{dy}{dx} = 8^x \cdot \ln(8)$$

Apply the rule to our function

Now, we just need to apply the rule from step 1 to our given function, $$y = 8^x$$. So, we have: $$\frac{dy}{dx} = 8^x \cdot \ln(8)$$ That's it! The derivative of the function $$y = 8^x$$ is $$\frac{dy}{dx} = 8^x \cdot \ln(8)$$.

Key concepts

Exponential Functions

Exponential functions are mathematical functions of the form \(y = a^x\) where \(a\) is a constant known as the base and \(x\) is the variable exponent. These functions are widespread in various fields because they represent growth or decay processes, such as population growth or radioactive decay. One of the key characteristics of exponential functions is their constant growth rate, meaning the rate of change of the function is proportional to the current value of the function itself.

To identify an exponential function, look for a variable exponent with a constant base. For example, in the function \(y = 8^x\), the base is 8 and it remains constant, while \(x\) varies. This creates a swiftly increasing or decreasing curve, depending on whether the base is greater or less than one. Understanding how to differentiate these functions helps in determining tangential slopes, optimizing problems, and analyzing data trends in predictions.

Exponential functions are foundational in calculus, making it crucial for you to understand their properties and behaviors.

Differentiation Rules

Differentiation is a core concept in calculus that allows us to find the rate of change of a function. With functions like \(y = a^x\), differentiation rules provide a systematic way to calculate derivatives efficiently without always reverting to first principles.

For exponential functions, the derivative can be found using a specific rule: for any function \(y = a^x\), the derivative \(\frac{dy}{dx}\) is calculated as \(a^x \cdot \ln(a)\). In this rule:

• \(a^x\) represents the original function.
• \(\ln(a)\) is the natural logarithm of the base \(a\).

The derivative gives us the function's rate of change concerning \(x\). For example, applying this rule, the derivative of \(y = 8^x\) is \(8^x \cdot \ln(8)\). This means for any value of \(x\), the slope of the tangent to the function curve at that point can be calculated using this derivative.

Mastering these differentiation rules simplifies solving complex calculus problems and is essential for dealing with exponential growth and decay scenarios.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is approximately 2.71828. This constant \(e\) is unique and crucial in mathematics because it frequently appears in calculations involving growth and decay, compounding, and in differential equations.

In the context of differentiating exponential functions, the natural logarithm plays a significant role. The derivative of an exponential function \(y = a^x\) relies on multiplying the original function \(a^x\) by \(\ln(a)\). This operation comes from the chain rule of differentiation as applied to exponential functions, which essentially tells us how the function's growth rate is scaled by the logarithm of the base.

• \(\ln(a)\) modifies the exponential rate of change depending on \(a\).
• Utilizing \(\ln\) makes derivatives of exponential functions manageable.

Understanding the behavior and properties of \(\ln\) helps explain why certain functions grow faster or slower and assists in optimizing processes involving exponential growth, such as finance, biology, or physics.|