Problem 14 - The Calculus with Analytic Geometry

Question

Find the derivative of the given function. $$ h(x)=\left(\frac{x+4}{2 x^{2}-5 x+6}\right)^{3} $$

Step-by-Step Solution

Answer

The derivative is \( h'(x) = \frac{-6(x+4)^2(x^2 + 7)}{(2x^2 - 5x + 6)^4} \).

Step 1: Recognize the Function

Given the function: \[h(x)=\left(\frac{x+4}{2x^{2}-5x+6}\right)^{3}\] This is a composite function with an outer function, \(u^3\), and an inner function, \( \left(\frac{x+4}{2x^{2}-5x+6}\right) \). Identify each part clearly before proceeding.

Step 2: Apply the Chain Rule

To find the derivative \( h'(x) \), use the chain rule: if \( h(x) = f(g(x)) \), then \( h'(x) = f'(g(x)) \, g'(x) \). For \( f(u)=u^3 \) and \( g(x)=\frac{x+4}{2x^{2}-5x+6} \), we need to find \( f'(u) \) and \( g'(x) \).

Step 3: Differentiate the Outer Function

Differentiate the outer function \( f(u)=u^3 \). Using the power rule, \( f'(u) = 3u^2 \). Substitute back \( u = \frac{x+4}{2x^{2}-5x+6} \).

Step 4: Differentiate the Inner Function

Now, find the derivative of the inner function \( g(x)=\frac{x+4}{2x^{2}-5x+6} \). Use the quotient rule: \[ g'(x) = \frac{(v(x)u'(x) - u(x)v'(x))}{v(x)^2} \]where \( u(x) = x+4 \) and \( v(x) = 2x^2 - 5x + 6 \).

Step 5: Compute Derivatives for Quotient Rule

Identify and compute necessary derivatives: \[ u(x) = x+4, \quad u'(x) = 1 \]\[ v(x) = 2x^2 - 5x + 6, \quad v'(x) = 4x - 5 \]

Step 6: Apply Quotient Rule

Substitute these into the quotient rule formula:\[ g'(x) = \frac{(2x^2 - 5x + 6)(1) - (x+4)(4x-5)}{(2x^2 - 5x + 6)^2} \]Simplify the numerator:\[ = \frac{2x^2 - 5x + 6 - 4x^2 + 5x - 20}{(2x^2 - 5x + 6)^2} \]\[ = \frac{-2x^2 - 14}{(2x^2 - 5x + 6)^2} \]\[ = \frac{-2(x^2 + 7)}{(2x^2 - 5x + 6)^2} \]

Step 7: Combine Results

The derivative of the outer function with respect to the inner function is:\[ 3 \left( \frac{x+4}{2x^2 - 5x + 6} \right)^2 \]Multiply this by the derivative of the inner function:\[ h'(x) = 3 \left( \frac{x+4}{2x^2 - 5x + 6} \right)^2 \cdot \frac{-2(x^2 + 7)}{(2x^2 - 5x + 6)^2} \]Simplify the final result:\[ h'(x) = \frac{-6(x+4)^2(x^2 + 7)}{(2x^2 - 5x + 6)^4} \]

Key Concepts

calculus

Calculus is a branch of mathematics that primarily deals with the concept of change. It includes two major areas: differentiation and integration. Differentiation focuses on how functions change, which is essentially about finding the slope of a function at any given point.
Integration, on the other hand, is about finding the area under the curve of a function.
Derivatives are a key part of calculus, often used to find the rate of change of variables in physics, engineering, economics, and many other fields.
This exercise focuses on differentiation, particularly using rules like the chain rule and the quotient rule to find the derivative of a more complex composite function.
The goal is to break down a complicated function into simpler parts to find how it changes with respect to its variable.

chain rule

The chain rule is a fundamental tool in calculus for finding the derivative of composite functions. When you have a function that is made up of two or more nested functions, the chain rule allows you to differentiate it step by step.
In this exercise, we have a function of the form \( h(x) = f(g(x)) \), where \( f(u) = u^3 \) and \( g(x) = \left(\frac{x+4}{2x^2 - 5x + 6}\right) \).
To apply the chain rule, follow these steps:

• First, differentiate the outer function \( f(u) = u^3 \). Using the power rule, this becomes \( f'(u) = 3u^2 \).
• Second, differentiate the inner function \( g(x) = \left(\frac{x+4}{2x^2 - 5x + 6}\right) \) using the quotient rule.

The chain rule ties together the differentiation of these nested functions to find the overall derivative of \( h(x)\).

quotient rule

The quotient rule is used in calculus to find the derivative of a function that is the ratio of two other functions. In our exercise, the inner function \( g(x) = \left(\frac{x+4}{2x^2 - 5x + 6}\right) \), requires the application of the quotient rule to differentiate.
The quotient rule states that if you have a function \( g(x) = \frac{u(x)}{v(x)} \), then its derivative is:
\[ g'(x) = \frac{(v(x)u'(x) - u(x)v'(x))}{v(x)^2} \]
Applying this to our inner function:

• Let \( u(x) = x+4 \) and \( v(x) = 2x^2 - 5x + 6\)
• First, find the derivatives \( u'(x) = 1 \) and \( v'(x) = 4x - 5 \)

Next, substitute these into the quotient rule formula:
\[ g'(x) = \frac{(2x^2 - 5x + 6)(1) - (x+4)(4x-5)}{(2x^2 - 5x + 6)^2} \]
Finally, simplify to get the derivative of the inner function. This step completes the differentiation using the quotient rule.

composite functions

Composite functions are functions that are formed by combining two or more functions where the output of one function becomes the input of another. We denote a composite function as \( h(x) = f(g(x)) \).
In our exercise, the function \( h(x) = \left(\frac{x+4}{2x^2 - 5x + 6}\right)^{3} \) is a composite function where \( f(u) = u^3 \) and \( g(x) = \left(\frac{x+4}{2x^2 - 5x + 6}\right) \).
When dealing with composite functions:

• First, identify the outer and inner functions clearly.
• Next, apply the chain rule to differentiate the outer function with respect to the inner function.
• Then, differentiate the inner function separately using appropriate rules like the quotient rule.

Combining these steps, you can find the overall derivative of \( h(x) \). This method simplifies the differentiation of complex functions by breaking them down into manageable parts.