91Ó°ÊÓ

Calculus is the branch of mathematics that studies continuous change. It has two main branches: differentiation and integration. Differentiation deals with the rate at which quantities change, while integration is about finding the total amount or value.

When solving calculus problems, you often encounter functions that describe how something changes over time or space. Calculus provides tools to deeply understand these changes by breaking down functions into manageable pieces.

In this exercise, we use the quotient rule, a differentiation technique to handle functions presented as the division of two other functions.
The quotient rule is a fundamental part of calculus and helps us differentiate more complex functions.

The chain rule is a crucial rule in calculus for differentiation of composite functions. A composite function is a function made up of other functions, such as \( h(g(x)) \).

To differentiate a composite function, the chain rule states: \[ \frac{dh}{dx} = \frac{dh}{dg} \times \frac{dg}{dx} \] For the given problem, we have \( v = (x + \sin x)^2 \), which can be considered as \( h(g(x)) \) where \( g(x) = x + \sin x \) and \( h(g) = g^2 \).

By applying the chain rule, we break down the problem into: \[ \frac{dv}{dx} = 2(x + \sin x) \times (1 + \cos x) \]This represents a straightforward way to handle more complicated differentiation tasks. Understanding this rule is critical for mastering calculus and makes handling intricate functions much more manageable.

Differentiation is the process of finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable.

In this exercise, our task is to find the derivative of \( y = \frac{x}{(x + \sin x)^2}\).

Differentiation involves applying various rules, such as the quotient rule, chain rule, and product rule.

Using the quotient rule, we differentiate the numerator \( x \) and the denominator \( (x + \sin x)^2 \) separately, then combine the results:
\[\frac{d}{dx} \bigg( \frac{u}{v} \bigg) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
In this way, differentiation helps us understand how \( y \) changes with respect to \( x \).

Simplification is a critical step in solving calculus problems, making the expression easier to read and work with.

After applying the quotient rule and chain rule, the expression for the derivative becomes complex. We simplify it by combining like terms and factoring.

For example, the numerator simplifies as follows:
\[ (x + \sin x)^2 - 2x(x + \sin x)(1 + \cos x) = (x + \sin x)( \sin x - x - 2x \cos x) \]

Cancelling out \( x + \sin x \) from the numerator and denominator, we have:
\[ \frac{ \sin x - x - 2x \cos x }{(x + \sin x)^3} \]

Each simplification step makes the final expression more straightforward to interpret and use.