Problem 40 - Calculus of a Single Variable

Question

Finding a Derivative In Exercises \(37-58\) , find the derivative of the function. (Hint: In some exercises, you may find it helpful to apply logarithmic properties before differentiating.) $$ y=6^{3 x-4} $$

Step-by-Step Solution

Answer

The derivative of the function \(y=6^{3x-4}\) is given by \( \frac{dy}{dx} = 3 \cdot 6^{3x-4} \ln 6\).

Step 1: Applying Logarithm Properties

Start by taking the natural logarithm on both sides of the equation: \[ \ln y = \ln (6^{3x - 4}) \]. Make use of the logarithmic property \( \ln a^b = b \ln a \) to expand the right hand side of the equation: \[ \ln y = (3x - 4) \ln 6 \] .

Step 2: Differentiation

Now differentiate the equation implicitly with respect to \(x\): \[ \frac{1}{y} \frac{dy}{dx} = 3 \ln 6 \]. Afterwards, solve for \( \frac{dy}{dx} \) to obtain: \[ \frac{dy}{dx} = 3y \ln 6 \].

Step 3: Replace y with Original Function

Lastly, replace \(y\) in the equation with the original function \(6^{3x-4}\) to get \[ \frac{dy}{dx} = 3 \cdot 6^{3x-4} \ln 6 \]. This is the derivative of the function.

Key Concepts

Logarithmic Properties

The use of logarithmic properties is an invaluable tool in calculus, especially when dealing with functions involving exponents. One of the most helpful properties is \( \ln a^b = b \ln a \). This property allows us to bring exponent terms down in front of the logarithm as a multiplier, making differentiation much simpler.
Let's see how this applies to the given exercises. Begin by taking the natural logarithm of both sides of the equation \( y=6^{3x-4} \). Apply the property to simplify the expression: \( \ln y = (3x - 4) \ln 6 \).
This transformation simplifies the differentiation process by allowing you to work with a linear expression instead of an exponential one. This step sets the stage for using implicit differentiation to find the derivative.

Implicit Differentiation

Implicit differentiation is a technique used when it is challenging to explicitly solve for one variable in terms of another beforehand. It becomes useful when dealing with equations where the dependent variable is part of a more complex expression.
For the equation \( \ln y = (3x - 4) \ln 6 \), treating \( y \) as a function of \( x \), we differentiate both sides with respect to \( x \). This process is where we apply implicit differentiation.

• The derivative of \( \ln y \) is \( \frac{1}{y} \frac{dy}{dx} \).
• The derivative of \((3x - 4)\ln 6\) is simply \(3 \ln 6\).

This provides the equation \( \frac{1}{y} \frac{dy}{dx} = 3 \ln 6 \). From this, multiplying both sides by \( y \) isolates \( \frac{dy}{dx} \), leading us to \( \frac{dy}{dx} = 3y \ln 6 \).

Exponential Functions

Exponential functions of the form \( a^{f(x)} \) appear commonly in both theoretical and applied math. Understanding their behavior is crucial when calculating derivatives. These functions grow or decay at a rate proportional to their current value.
In our specific example, \( y = 6^{3x-4} \), the function exhibits exponential growth. Calculating derivatives of exponential functions often involves retaining the original function form. Once you find the derivative implicitly, replace \( y \) with \( 6^{3x-4} \).
This results in \( \frac{dy}{dx} = 3 \cdot 6^{3x-4} \ln 6 \). Here, the derivative is expressed in terms of the initial exponential function, scaled by the factor \( 3 \ln 6 \). This solution highlights how exponential operations impact the rate of change in functions.