Problem 54 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The derivative is \( \frac{-3\sin(3x)(2x^3 - x) - (1 + \cos(3x))(6x^2 - 1)}{(2x^3 - x)^2} \).

Step 1: Identify the Function Structure

The function is a quotient of two functions: the numerator \( 1 + \cos(3x) \) and the denominator \( 2x^3 - x \). To find the derivative, we will use the quotient rule.

Step 2: Apply the Quotient Rule

The quotient rule for derivatives states that the derivative of \( \frac{u}{v} \) is given by \( \frac{u'v - uv'}{v^2} \). Here, \( u = 1+\cos(3x) \) and \( v = 2x^3-x \). We need to find \( u' \) and \( v' \).

Step 3: Derive the Numerator

The derivative of the numerator \( u = 1 + \cos(3x) \) is \( u' = -3\sin(3x) \) because the derivative of constant is 0 and the derivative of \( \cos(3x) \) is \( -3\sin(3x) \) (using the chain rule).

Step 4: Derive the Denominator

The derivative of the denominator \( v = 2x^3 - x \) is \( v' = 6x^2 - 1 \).

Step 5: Substitute Derivatives into the Quotient Rule

Substitute \( u', v, u, \) and \( v' \) into the quotient rule formula: \( \frac{(-3\sin(3x))(2x^3 - x) - (1 + \cos(3x))(6x^2 - 1)}{(2x^3 - x)^2} \).

Step 6: Simplify the Expression

Simplify the expression in the numerator by distributing and combining like terms. This step may take some time, as it involves careful algebra.

Key Concepts

Quotient Rule

In calculus, the quotient rule is a handy tool when you want to differentiate a function that is the quotient of two other functions. Essentially, the quotient rule is a method for finding the derivative of a function written as a fraction, where the top and bottom are both differentiable functions.

The formula is straightforward but involves a few steps. If you have a function expressed as \( \frac{u}{v} \), where both \( u \) and \( v \) are functions of some variable (like \( x \)), the derivative \( \left( \frac{d}{dx} \right) \) is computed as:

\[\frac{u'v - uv'}{v^2}\]Here:

• \( u' \) is the derivative of \( u \)
• \( v' \) is the derivative of \( v \)

The process involves first finding these derivatives and then plugging them into the quotient rule formula to get the derivative of the entire function. It's essential to perform the algebraic simplifications carefully at the end to ensure the solution is in its simplest form.

Things to remember:

• The denominator \( v^2 \) means you will never have a zero in the denominator, as long as \( v \) itself is not zero.
• Always simplify the algebraic expression as much as possible. This can help avoid errors and make your solution clearer.

Chain Rule

The chain rule is another fundamental calculus concept that allows you to differentiate more complex functions, especially those that involve "functions inside functions."

When you encounter a composite function, like \( \cos(3x) \) in the problem, the chain rule becomes necessary. A composite function can be thought of as \( f(g(x)) \), where you have an outer function \( f \) acting on an inner function \( g(x) \).

To apply the chain rule, you multiply the derivative of the outer function evaluated at the inner function by the derivative of the inner function itself. Mathematically, if you have \( y = f(g(x)) \), the derivative \( \frac{dy}{dx} \) is:

\[f'(g(x)) \cdot g'(x)\]In our exercise, for \( u = 1 + \cos(3x) \), we use the chain rule to find \( u' \).

• The derivative of \( \cos \) is \(-\sin \), so \( \frac{d}{d(3x)} \cos(3x) = -\sin(3x) \).
• The derivative of the inner function \( 3x \) is \( 3 \).

Therefore, multiplying these gives \( -3\sin(3x) \), which is the derivative of \( \cos(3x) \) with respect to \( x \) using the chain rule.

Calculus Problems

Working through calculus problems can seem daunting, but breaking them down into smaller parts makes them manageable. Let's consider steps to tackle derivatives effectively:

1. **Understand the function**: Identify the structure. Is it a simple polynomial, or do you need the chain or quotient rule? 2. **Differentiate step by step**:

• Apply the appropriate rule. For fractions, use the quotient rule.
• Use the chain rule for nested functions.
• Identify constants and simple terms that do not change when differentiated.

3. **Algebraic Simplification**: After differentiation:

• Expand terms if necessary, combining like terms.
• Simplify the final expression to get your answer in the simplest form.

4. **Check your work**: Re-examine each step to ensure correctness, especially the algebraic simplification parts. This diligence keeps mistakes at bay.

Grasping these strategies and understanding how rules apply to various situations develop essential calculus skills which are valuable in both academics and real-world applications.