Problem 53 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The derivative of the function is \( \frac{2\cos(2x) + 2x^2\cos(2x) - 2x\sin(2x)}{(1 + x^2)^2} \).

Step 1: Identify the Necessary Derivative Rule

To differentiate the function \( f(x) = \frac{\sin(2x)}{1+x^2} \), we need to use the quotient rule for derivatives. The quotient rule states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{v(u') - u(v')}{v^2} \), where \( u \) and \( v \) are functions of \( x \).

Step 2: Define Functions for Quotient Rule

Let \( u = \sin(2x) \) and \( v = 1 + x^2 \). Our task is to find the derivatives \( u' \) and \( v' \).

Step 3: Differentiate the Numerator \( u \)

For \( u = \sin(2x) \), use the chain rule to find \( u' \). The derivative of \( \sin(2x) \) is \( \cos(2x) \cdot 2 = 2\cos(2x) \).

Step 4: Differentiate the Denominator \( v \)

For \( v = 1 + x^2 \), differentiate with respect to \( x \) to find \( v' \). The derivative of \( 1 + x^2 \) is \( 0 + 2x = 2x \).

Step 5: Apply the Quotient Rule

Substitute \( u = \sin(2x) \), \( u' = 2\cos(2x) \), \( v = 1 + x^2 \), and \( v' = 2x \) into the quotient rule:\[\frac{d}{dx}\left(\frac{\sin(2x)}{1 + x^2}\right) = \frac{(1 + x^2)(2\cos(2x)) - (\sin(2x))(2x)}{(1 + x^2)^2}\]

Step 6: Simplify the Result

Simplify the expression: \( (1 + x^2)(2\cos(2x)) - (\sin(2x))(2x) = 2\cos(2x) + 2x^2\cos(2x) - 2x\sin(2x) \).Thus, the final derivative is:\[\frac{2\cos(2x) + 2x^2\cos(2x) - 2x\sin(2x)}{(1 + x^2)^2}\]

Key Concepts

Quotient Rule

In calculus, the Quotient Rule is a method for finding the derivative of a quotient of two functions. It's particularly helpful when you have one function divided by another. When trying to differentiate a function expressed as \( \frac{u}{v} \), the rule states that the derivative is \( \frac{v(u') - u(v')}{v^2} \). Here, \( u \) and \( v \) are functions of the same variable, and \( u' \) and \( v' \) represent the derivatives of these functions, respectively.
The Quotient Rule arises as an extension of the Product Rule and is essential for ensuring precise calculations in calculus. When applying it, keep this sequence in mind:

• First, identify \( u \) and \( v \) from your function.
• Calculate the derivatives \( u' \) and \( v' \).
• Substitute these into the quotient rule formula.

With practice, spotting when to use the Quotient Rule and carrying out these steps will become second nature.

Chain Rule

The Chain Rule is a fundamental technique used for differentiating composite functions. It allows us to find the derivative of a function that is nested within another function, which is termed as a composite function. In mathematical terms, if you have a function \( y = f(g(x)) \), the Chain Rule states that the derivative \( y' \) is given by \( f'(g(x)) \cdot g'(x) \).
When applying the Chain Rule, identify the 'outer function' and the 'inner function.' The outer function, \( f(g(x)) \), is differentiated first, then multiply the result by the derivative of the inner function \( g(x) \). Here’s a step-by-step breakdown:

• Find the derivative of the outer function, holding the inner function constant.
• Determine the derivative of the inner function.
• Multiply these derivatives together to get the final result.

This method helps simplify complex functions into manageable parts, and is crucial for correctly solving various calculus problems.

Trigonometric Functions

Trigonometric functions, such as \( \sin(x), \cos(x), \text{ and } \tan(x) \), are fundamental in mathematics, especially in calculus. They occasionally appear within derivative problems, such as the one at hand. Differentiating these functions requires knowledge of specific rules. For instance:

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

When trigonometric functions are combined with other types of functions, they often involve using both the Chain Rule and the Quotient Rule, as seen in this exercise. Remember that recognizing patterns in trigonometric identities can also assist in simplifying expressions and making derivatives more manageable. Using these rules and strategies will help you effectively tackle functions that include trigonometric elements.