Problem 43 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the derivative of the function. $$ h(x)=4^{2 x-3} $$

Step-by-Step Solution

Answer

The derivative of the function \( h(x) = 4^{2x-3} \) is \( h'(x) = 2 \cdot 4^{2x-3} \cdot ln(4) \).

Step 1: Identify the function

First, identify the function which is \( h(x) = 4^{2x-3} \). It's a function of a function which can be written as \( u^{v} \) where \( u = 4 \) and \( v = 2x - 3 \).

Step 2: Differentiate the function using the power rule

For a function in the form of \( u^{v} \), its derivative can be found using the formula: \( (u^{v})' = u^{v} \cdot ln(u) \cdot v' \). Now, implement this formula to differentiate \( h(x) \), which gives \( h'(x) = 4^{2x-3} \cdot ln(4) \cdot (2x - 3)' \). The derivative of \( 2x - 3 \) is 2, so the formula results in \( h'(x) = 4^{2x-3} \cdot ln(4) \cdot 2 \).

Step 3: Simplify the function

Now, simplify the function to get the final derivative of the function. \( h'(x) = 2 \cdot 4^{2x-3} \cdot ln(4) \).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus that helps us find the derivative of composite functions. A composite function is a function that is made up of two or more functions. For example, if you have a function like \( h(x) = f(g(x)) \), it means you're taking one function \( g(x) \), and putting it into another function \( f \).
To apply the chain rule:

• Differentiation starts from the outside function. Think of peeling an onion, where you start from the outer layer, working your way inwards.
• The chain rule tells us that the derivative of \( h(x) = f(g(x)) \) is \( h'(x) = f'(g(x)) \cdot g'(x) \).

In simpler terms, you first take the derivative of the outer function \( f \), treating the inner function \( g(x) \) as a variable. Then multiply it by the derivative of the inner function \( g(x) \). This process is all about ensuring that we account for how each piece of our composite function changes.

Exponential Functions

Exponential functions are functions that have a constant base raised to a variable exponent, written as \( b^x \), where \( b \) is a constant and \( x \) is the exponent. A key aspect of these functions is their rapid growth or decay, making them very different from polynomial functions.
For exponential functions with a base different from \( e \) (like \( 4^{2x-3} \)), the derivative involves using the natural logarithm, \( ln(b) \), of the base. The beauty of exponential derivatives is in their simplicity:

• The derivative of \( b^x \) with respect to \( x \) is \( b^x \cdot ln(b) \).
• When finding the derivative of an exponential function that is more complex, like having a function in the exponent (e.g., \( 4^{2x-3} \)), you'll use both the exponential derivative rule and the chain rule.

This combination ensures we correctly account for changes in both the exponent and the base as we're differentiating.

Power Rule

The power rule is a shortcut to finding the derivative of expressions with a power, typically in the form of \( x^n \). It’s direct and simplifies processes where we raise variables by a constant exponent.
The power rule can be applied as follows:

• For a function \( x^n \), the derivative \( \frac{d}{dx}x^n \) equals \( nx^{n-1} \).

However, in the scenario of \( 4^{2x-3} \), this rule doesn't directly apply because we’re dealing with an exponential function, not a polynomial power. Still, understanding the power rule helps in the "peeling" process involved when combining it with other rules such as the chain rule or dealing with derivatives of more complex functions.
It's often utilized in parts wherever you have just a plain power function segment within different types of composite or complex derivatives.