Problem 24 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=-3 \cos ^{2}\left(3 x^{2}-4\right) $$

Step-by-Step Solution

Answer

The derivative is \(36x \cos(3x^2 - 4) \sin(3x^2 - 4)\).

Step 1: Identify the Overall Structure

The function given is a composition of functions: the squaring function, the cosine function, and the polynomial inside the cosine. We will apply the chain rule for derivatives.

Step 2: Differentiate the Outer Function

The outer function is \[-3(u)^2\] where \(u = \cos(3x^2 - 4)\). Use the power rule to differentiate, giving \[\frac{d}{du} [-3u^2] = -6u\].

Step 3: Differentiate the Middle Function

The middle function is \(\cos(3x^2 - 4)\). The derivative of \(\cos v\) with respect to \(v\) is \[-\sin v\]. So, the derivative with reference to \(u\) is \[-\sin(3x^2 - 4)\].

Step 4: Differentiate the Inner Function

The inner function is the polynomial \(3x^2 - 4\). The derivative of a polynomial \(ax^n\) is \[anx^{n-1}\]. Thus, the derivative of \(3x^2 - 4\) is \[6x\].

Step 5: Apply the Chain Rule

Combine the derivatives using the chain rule. Multiply the derivatives of each nested function form:1. \(-6 u\) from Step 2.2. \(-\sin(3x^2 - 4)\) from Step 3.3. \(6x\) from Step 4.Give the combined derivative:\[(-6)(\cos(3x^2 - 4))(-\sin(3x^2 - 4))(6x) = 36x \cos(3x^2 - 4) \sin(3x^2 - 4).\]

Step 6: Simplify the Expression

The product of \(-6\), \(-1\), and \(6\) gives \(36\). Thus, the full derivative simplifies to:\[36x \cos(3x^2 - 4) \sin(3x^2 - 4).\]

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. Composite functions are functions within other functions, like nesting one inside another. The chain rule essentially helps in taking the derivative of these nested, composite functions.
To use the chain rule, follow these steps:

• Identify the outer function and the inner function. For example, if you have a function such as \( f(g(x)) \), then \( f \) is the outer function and \( g(x) \) is the inner function.
• Differentiate the outer function first, keeping the inner function intact. This means you treat the inner part as a single unit or variable.
• Next, multiply by the derivative of the inner function. This accounts for how the inner function is changing.

With these steps, you can separate the differentiation task into manageable pieces. This makes finding the derivative of complex functions much more attainable.

Power Rule

The power rule is one of the simplest and most commonly used rules for differentiation. It's applied when taking the derivative of a variable raised to a power. The general format is \( f(x) = x^n \), where \( n \) is any real number.
To differentiate using the power rule:

• Bring the exponent down in front of the variable. This means multiplying the entire term by the exponent.
• Subtract one from the exponent to find the new power of the variable.

For example, if you have \( f(x) = x^3 \), using the power rule gives you \( f'(x) = 3x^{2} \). This straightforward approach is effective for polynomials as well as for each component of more complicated functions, such as those involved in the chain rule applications.

Trigonometric Functions

Understanding derivatives of trigonometric functions is essential for solving many calculus problems. Common trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each has its specific rule for differentiation:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).
• The derivative of \( \tan(x) \) is \( \sec^2(x) \).

In any composite function where a trigonometric function is involved, keep in mind these basic derivatives. For example, if working with \( \cos(3x^2 - 4) \), the derivative replaces \( \cos \) with \(-\sin\), and you still need to continue applying the chain rule to account for the inside function of \( 3x^2 - 4 \).

Composite Function Differentiation

Differentiating composite functions involves understanding how to manage functions embedded within each other and requires the chain rule. A composite function, by definition, is a function that applies one function to the result of another, such as \( h(x) = f(g(x)) \).
Steps to differentiate a composite function:

• Use the chain rule to first differentiate the outer function, treating inner functions as singular variables.
• Continue to differentiate each layer of function inwards, multiplying derivatives as you go along.
• Make sure to simplify at the end to find the cleanest derivative expression possible.

This process can feel complex at first, but with practice, disassembling and differentiating each layer of the function accurately becomes intuitive. The key takeaway is to approach each layer of the composite function like peeling an onion, differentiating piece by piece, following the nestled instructions of the chain rule.