Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\). $$ y=\frac{1}{\left(3 x^{2}+x-3\right)^{9}} $$

Step-by-Step Solution

Answer

\( D_x y = \frac{-9(6x + 1)}{(3x^2 + x - 3)^{10}} \).

Step 1: Rewrite the Function

First, rewrite the given function in terms of a power of a function: \( y = (3x^2 + x - 3)^{-9} \). This facilitates the application of the chain rule for differentiation.

Step 2: Apply the Chain Rule

The chain rule for differentiation states that if you have a composite function \( (u(x))^n \), the derivative is \( nu'(x)(u(x))^{n-1} \). Here, let \( u(x) = 3x^2 + x - 3 \) and \( n = -9 \). Therefore, \( \frac{d}{dx}y = -9u'(x)(u(x))^{-10} \).

Step 3: Find u'(x)

Differentiate \( u(x) = 3x^2 + x - 3 \). The derivative \( u'(x) = 6x + 1 \).

Step 4: Substitute Back Into the Chain Rule

Substitute \( u(x) \) and \( u'(x) \) into the derivative expression from Step 2: \( D_x y = -9(6x + 1)(3x^2 + x - 3)^{-10} \).

Step 5: Simplify the Result

The expression from Step 4 is already fairly simple but can be written concisely if necessary. The final derivative is: \( D_x y = \frac{-9(6x + 1)}{(3x^2 + x - 3)^{10}} \).

Key Concepts

Chain rule

In calculus differentiation, the chain rule is an essential tool when working with composite functions like \( y = (3x^2 + x - 3)^{-9} \). The chain rule allows you to differentiate a composition of functions efficiently. When you encounter a function written as \( (u(x))^n \), where \( u(x) \) itself is a function of \( x \), the chain rule states that the derivative is:

• \( n \, u'(x) \, (u(x))^{n-1} \)

In our example, the outer function is the power \( n = -9 \), and the inner function is \( u(x) = 3x^2 + x - 3 \). By identifying these components, you can apply the chain rule by:

• Calculating the derivative of the outer function with respect to the inner function.
• Multiplying it by the derivative of the inner function \( u'(x) \).

This method is particularly useful for functions with intricate nesting, ensuring accurate and swift differentiation.

Composite function

A composite function involves combining two or more functions. In our specific example, \( y = (3x^2 + x - 3)^{-9} \), the function \( y \) is the combination of two simpler functions: the polynomial \( u(x) = 3x^2 + x - 3 \) and the power function applied to it.The process of differentiation becomes more complex with composite functions, but it is also more structured:

• Identify the inner function \( u(x) \).
• Apply the derivative rules specifically catered to compositions.

For students, realizing a problem involves a composite function helps simplify finding the derivative. Breaking it down into each function component simplifies applying the chain rule.

Power rule

The power rule in differentiation is a fundamental rule often combined with other techniques like the chain rule. It applies to functions in the form of \( x^n \), stating that the derivative of \( x^n \) is \( n \, x^{n-1} \).In the given function \( y = (3x^2 + x - 3)^{-9} \), once rewritten as \((u(x))^n\), the power of \( -9 \) applies:

• Bring down the exponent \(-9\) and multiply it by the function.
• Reduce the exponent by one to \(-10\).

This step prepares the function for further processing using the chain rule by accounting for the power law's influence on the overall derivative.

Derivative simplification

After finding the derivative using the chain and power rules, simplifying the expression can make it more interpretable and useful. In our case, the derivative calculating steps yielded an expression:\[ D_x y = -9(6x + 1)(3x^2 + x - 3)^{-10} \]By rewriting this as:

• \( D_x y = \frac{-9(6x + 1)}{(3x^2 + x - 3)^{10}} \)

This transformation converts the power expression into a fractional format, which is often easier to handle in further analysis.Simplifying helps reduce computational complexity and improve the clarity of the final expression. This is crucial in solving calculus problems and preparing for subsequent operations.