Problem 16 - Brief Applied Calculus

Question

Use the Generalized Power Rule to find the derivative of each function. $$ h(z)=\left(5 z^{2}+3 z-1\right)^{3} $$

Step-by-Step Solution

Answer

The derivative is \( h'(z) = 3(5z^2 + 3z - 1)^2 (10z + 3) \).

Step 1: Identify the Outer and Inner Functions

The function given is a composite function where the outer function is \( u^3 \) and the inner function is \( u = 5z^2 + 3z - 1 \). We need to apply the chain rule to find the derivative, which involves differentiating the outer function and then multiplying it by the derivative of the inner function.

Step 2: Apply the Generalized Power Rule

The Generalized Power Rule for differentiation states that if you have a function \( (u(x))^n \), then the derivative is \( n \cdot (u(x))^{n-1} \cdot u'(x) \). For our function, \( n = 3 \), so we get: \[ d/dz[(5z^2 + 3z - 1)^3] = 3(5z^2 + 3z - 1)^{2} \times \frac{d}{dz}(5z^2 + 3z - 1). \]

Step 3: Differentiate the Inner Function

Now, find the derivative of the inner function \( u = 5z^2 + 3z - 1 \): \[ \frac{d}{dz}(5z^2 + 3z - 1) = 10z + 3. \]

Step 4: Combine the Derivatives

Substitute the derivative of the inner function found in Step 3 into the equation from Step 2: \[ h'(z) = 3(5z^2 + 3z - 1)^{2} \times (10z + 3). \]

Step 5: Simplify (If Needed)

Although the derivative can be left in this form, you can expand it for simplicity or application. However, the expression \( h'(z) = 3(5z^2 + 3z - 1)^2 (10z + 3) \) is already the simplified derivative using the Generalized Power Rule.

Key Concepts

Chain Rule

The Chain Rule is a powerful tool in calculus used to differentiate composite functions. A composite function is essentially a combination of two or more functions, where the output of one function becomes the input of another. Consider the function given in the problem. It is written as \( h(z) = (5z^2 + 3z - 1)^3 \), which can be seen as a function within a function. To differentiate this, you use the Chain Rule.

• First, identify the outer function. In this instance, it is \( u(z)^3 \).
• The inner function is \( u(z) = 5z^2 + 3z - 1 \).
• The Chain Rule states that to find the derivative of a composite function, you differentiate the outer function with respect to the inner function, and then multiply by the derivative of the inner function.

This approach makes handling complicated functions straightforward, by breaking them down into manageable parts. Applying this principle simplifies the process of differentiation significantly.

Generalized Power Rule

The Generalized Power Rule extends the basic power rule used for simple functions, allowing you to differentiate more complex expressions. When dealing with expressions like \( (u(x))^n \), where \( u(x) \) is a function and \( n \) is an exponent, the Generalized Power Rule provides a formula to calculate the derivative.

• Specifically, the derivative is calculated as: \( n \cdot (u(x))^{n-1} \cdot u'(x) \).
• This rule is essentially the Chain Rule combined with the standard power rule for differentiation.

In our given problem, we directly apply this rule:

• Set \( n = 3 \) since \( (5z^2 + 3z - 1) \) is raised to the third power.
• The general approach involves differentiating the power expression while keeping in mind the derivative of the inner content.

This makes it a fundamental tool for calculating derivatives of functions with complex power terms.

Composite Functions

Composite functions appear frequently in calculus, typically when one function is nestled inside another. To handle these effectively, it's crucial to identify each part distinctly.
Consider the function we are working with: \( h(z) = (5z^2 + 3z - 1)^3 \). Here you're dealing with two distinct functions.

• The outer function is \( x^3 \), where \( x = 5z^2 + 3z - 1 \).
• The inner function is clearly \( 5z^2 + 3z - 1 \).

These need to be differentiated separately before applying solutions like the Chain and Generalized Power Rules. Understanding composite functions and their derivatives is essential for calculus, as it allows mathematicians to tackle more intricate problems.