Problem 53 - Calculus for Biology and Medicine

Question

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

Step 1: Identify the Rule to Use

The function is given as a quotient of two functions; therefore, the Quotient Rule is applicable. According to the Quotient Rule, if \(y = \frac{u}{v}\), then the derivative \(\frac{dy}{dx}\) is given by \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}\).

Step 2: Define \(u\) and \(v\)

In this function, \(u = \sin(2x)\) and \(v = 1 + x^2\). We are required to find their derivatives, \(\frac{du}{dx}\) and \(\frac{dv}{dx}\).

Step 3: Differentiate \(u = \sin(2x)\)

To find \(\frac{du}{dx}\), apply the Chain Rule. The derivative of \(\sin(2x)\) is \(\cos(2x)\cdot 2\), which simplifies to \(2\cos(2x)\).

Step 4: Differentiate \(v = 1 + x^2\)

The derivative of a constant is zero and the derivative of \(x^2\) is \(2x\). Therefore, \(\frac{dv}{dx} = 2x\).

Step 5: Apply the Quotient Rule

Using the derivatives found in Steps 3 and 4 in the Quotient Rule formula, we have:\[\frac{d}{dx}\left(\frac{\sin(2x)}{1+x^2}\right) = \frac{(1+x^2) \cdot 2\cos(2x) - \sin(2x) \cdot 2x}{(1+x^2)^2}\]

Step 6: Simplify the Expression

Simplify the expression obtained in Step 5:\[\frac{d}{dx}\left(\frac{\sin(2x)}{1+x^2}\right) = \frac{2\cos(2x)(1 + x^2) - 2x\sin(2x)}{(1+x^2)^2}\]This is a simplification of the expression found by applying the Quotient Rule.

Key Concepts

Quotient Rule

The Quotient Rule is a handy tool in calculus used for finding the derivative of a function that can be expressed as the division of two differentiable functions. Imagine you have a function presented as a fraction where you have a top part (numerator) and a bottom part (denominator). The Quotient Rule helps in simplifying the process of finding the derivative of this fraction. The formula for the Quotient Rule is:

• \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \)

Here, \(u\) and \(v\) are functions of \(x\), and \(\frac{du}{dx}\) and \(\frac{dv}{dx}\) are their respective derivatives. It is important to note that while the rules are straightforward, careful attention must be given to the order of operations and simplification steps to ensure the correctness of the final derivative expression.In the given problem, \(u = \sin(2x)\) and \(v = 1 + x^2\) are identified as the numerator and denominator, respectively. Using the Quotient Rule allowed us to correctly take the derivative of this function by neatly packaging the separate derivations into a formula.

Chain Rule

The Chain Rule in calculus is a critical technique used for differentiating composite functions. A composite function is essentially a function that applies one function to the results of another. Think of it as a function inside another function, like layers in an onion. The Chain Rule helps peel these layers in a systematic way.The formula for the Chain Rule is:

• \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)

Here, \(g(x)\) is the inner function and \(f(g(x))\) is the outer function. The rule tells us to first differentiate the outer function at the inner function, and then multiply by the derivative of the inner function.In our example, in order to find the derivative of \(u = \sin(2x)\), we use the Chain Rule. We recognize \(2x\) as the inner function and \(\sin(\cdot)\) as the outer function. Differentiating \(\sin(\cdot)\) gives us \(\cos(\cdot)\) and multiplying by the derivative of \(2x\) gives \(2\cos(2x)\). This is a great example of how composite functions can be unraveled using the Chain Rule.

Derivative

In calculus, a derivative is essentially a way to measure how a function changes as its input changes. Think of it as the rate at which one quantity changes with respect to another. This is particularly useful for understanding motion, growth, and trends in various fields of study.The most basic form of a derivative, often denoted as \(\frac{dy}{dx}\), represents the slope of the tangent line to the function at any given point. A derivative summarizes the instantaneous rate of change, like capturing the speed of a car at a specific moment in time.In the given exercise, we aimed to find the derivative of \(f(x) = \frac{\sin(2x)}{1+x^2}\). The process involved several steps of differentiation, using both the Quotient Rule and the Chain Rule. By applying these differentiation rules effectively, the final derivative was obtained as:

• \( \frac{d}{dx} \left( \frac{\sin(2x)}{1+x^2} \right) = \frac{2\cos(2x)(1 + x^2) - 2x\sin(2x)}{(1+x^2)^2} \)

This complex expression highlights how derivatives can capture intricate relationships by peeling away at the layers of compounded functions.