Problem 42 - Calculus and its applications

Question

Differentiate each function $$ f(x)=\frac{(2 x+3)^{4}}{(3 x-2)^{5}} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{(2x+3)^3(-6x - 61)}{(3x-2)^{6}} \).

Step 1: Identify the Quotient Rule

The function given is a quotient of two functions: the numerator \((2x+3)^4\) and the denominator \((3x - 2)^5\). We will use the quotient rule for differentiation, which is: \( \frac{h(x)}{g(x)} \rightarrow \frac{h'(x)g(x) - h(x)g'(x)}{[g(x)]^2} \).

Step 2: Differentiate the Numerator

The numerator is \(h(x) = (2x + 3)^4\). We will use the chain rule to differentiate it. Let \(u = 2x + 3\), then \(h(x) = u^4\), and \(h'(x) = 4u^3 \cdot u'\) where \(u' = 2\). Therefore, \(h'(x) = 4(2x+3)^3 \cdot 2 = 8(2x+3)^3\).

Step 3: Differentiate the Denominator

The denominator is \(g(x) = (3x - 2)^5\). Again, using the chain rule: let \(v = 3x - 2\), then \(g(x) = v^5\), and \(g'(x) = 5v^4 \cdot v'\) where \(v' = 3\). Thus, \(g'(x) = 5(3x-2)^4 \cdot 3 = 15(3x-2)^4\).

Step 4: Apply the Quotient Rule

Substitute \(h'(x)\), \(h(x)\), \(g'(x)\), and \(g(x)\) into the quotient rule formula: \[\frac{h'(x)g(x) - h(x)g'(x)}{[g(x)]^2} = \frac{8(2x+3)^3 (3x-2)^5 - (2x+3)^4 15(3x-2)^4}{[(3x-2)^5]^2}\].

Step 5: Simplify the Expression

Simplify the expression obtained in Step 4. Factor out the common terms: \([(2x+3)^3(3x-2)^4]\) is common. This gives: \[(2x+3)^3(3x-2)^4\left(8(3x-2) - 15(2x+3)\right)\]. Simplify inside the brackets: \[8(3x-2) = 24x - 16\] and \[15(2x+3) = 30x + 45\], thus the difference is \[(24x - 16 - 30x - 45) = -6x - 61\].

Step 6: Final Simplified Answer

The final differentiated expression is: \[\frac{(2x+3)^3(3x-2)^4(-6x - 61)}{(3x-2)^{10}}\]. To simplify further, cancel \((3x - 2)^4\) from the numerator and denominator: \[\frac{(2x+3)^3(-6x - 61)}{(3x-2)^{6}}\].

Key Concepts

Quotient rule

In calculus, the quotient rule is essential for differentiating functions that are the ratio of two other functions. If you have a function \(f(x) = \frac{h(x)}{g(x)}\), the quotient rule states that to find \(f'(x)\), you use the formula: \[f'(x) = \frac{h'(x)g(x) - h(x)g'(x)}{[g(x)]^2}\] This rule allows you to systematically derive the derivative of the quotient by utilising the derivatives of the numerator and the denominator separately.
In practice, this involves several clear steps:

• Differentiate the numerator, \(h(x)\), to find \(h'(x)\).
• Differentiate the denominator, \(g(x)\), to find \(g'(x)\).
• Apply these derivatives into the quotient rule formula.

Such application ensures that the differentiation is handled accurately, especially when the functions involved are complex. This method is generally straightforward, but attention to detail is required to ensure each function and its derivative are accurately represented before substitution.

Chain rule

The chain rule is a fundamental principle in calculus for differentiating composite functions. A composite function is, essentially, a function within another function, such as \(y = (2x + 3)^4\). Here, the chain rule allows us to differentiate by breaking down the computation into manageable parts.
The chain rule involves the following steps:

• Identify the outer function and the inner function. In our example, the outer function is \(u^4\) and the inner function is \(u = 2x + 3\).
• Differentiate the outer function with respect to the inner function, and then multiply it by the derivative of the inner function.
• For \(h(x) = (2x + 3)^4\), the derivative becomes \(h'(x) = 4(2x + 3)^3 imes 2\), after applying the chain rule.

The application of the chain rule is crucial when dealing with complex functions, as it enables differentiation to proceed through layers of functions systematically.

Simplification in calculus

Simplification is a vital part of calculus, especially after applying rules such as the quotient and chain rules. It often transforms complex expressions into more manageable forms.
When simplifying, always look for common factors that can be cancelled or terms that can be combined. In our example, the expression after applying the quotient rule results in:

• Identifying common factors such as \((2x+3)^3(3x-2)^4\) in the numerator.
• Factor these common elements out to simplify the expression.
• Simplify the resultant terms within brackets. Doing so reduced our original expression significantly.

Ultimately, simplification aims to present the derivative in its most concise form while preserving accuracy. Ensuring that needless complexity is removed makes further calculus operations or evaluations easier and clearer.