Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-20, find \(D_{x} y\). $$ y=\sin \left(x^{2}+x\right) $$

Step-by-Step Solution

Answer

\( D_{x} y = \cos(x^2 + x) \cdot (2x + 1) \)

Step 1: Identify the Function

We need to differentiate the function \( y = \sin(x^2 + x) \) with respect to \( x \). The function \( y \) is composed of an outer function, \( \sin(u) \), where \( u = x^2 + x \).

Step 2: Use the Chain Rule

The chain rule states that \( D_{x} y = \frac{dy}{du} \cdot \frac{du}{dx} \). In this situation, \( \frac{dy}{du} = \cos(u) \) for the outer function \( \sin(u) \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( u = x^2 + x \) with respect to \( x \):\[\frac{du}{dx} = \frac{d}{dx}(x^2 + x) = 2x + 1.\]

Step 4: Combine Using the Chain Rule

Now combine the derivatives from Step 2 and Step 3:\[ D_{x} y = \cos(u) \cdot \frac{du}{dx} = \cos(x^2 + x) \cdot (2x + 1). \]

Step 5: Final Solution

Thus, the derivative of the function \( y = \sin(x^2 + x) \) with respect to \( x \) is:\[ D_{x} y = \cos(x^2 + x) \cdot (2x + 1). \]

Key Concepts

Chain Rule

In calculus, the chain rule is a powerful technique used to differentiate composite functions. A composite function is when one function is nestled inside another, like the classic example of \(y = \sin(x^2 + x)\). Here, we see an inner function \(u = x^2 + x\), wrapped by \(\sin(u)\).

The chain rule helps us by providing a systematic way to find the derivative of such functions. It states that if you have a composite function, the derivative can be found by differentiating the outer function and then multiplying it by the derivative of the inner function.

Using the chain rule makes tackling complex functions considerably easier.

• Differentiating the outer function: \(\sin(u)\) becomes \(\cos(u)\).
• Differentiating the inner function: \(u = x^2 + x\) differentiates to \(2x + 1\).
• Combine the results: \(\cos(x^2 + x) \cdot (2x + 1)\).

Give it a try next time you encounter a composite function!

Trigonometric Functions

Trigonometric functions are a crucial part of both calculus and precalculus mathematics. They arise naturally in various problems, especially those involving periodic phenomena such as waves or rotational motions. These functions include sine, cosine, and tangent, among others.

In calculus, differentiating trigonometric functions frequently emerges when tackling problems. For example, the derivative of \(\sin(x)\) is \(\cos(x)\). This relationship allows us to find changes in the shape and direction of the wave represented by the sinusoidal function.

Understanding how these functions behave is key. Their graphs and periodic properties drive many applications, from engineering tasks to physical sciences.

• The derivative \(\sin\) converts to \(\cos\), posing a unique property.
• Each trigonometric derivative carries a transformation similar to \(\sin\) to \(\cos\).

Recognize patterns in trigonometric derivatives, as they reveal much about the original function's behavior.

Derivatives

A derivative refers to the rate at which a function changes, serving as one of the fundamental concepts in calculus. It describes how a function's output varies as its input alters slightly. Differentiating a function means finding this rate of change.

In our exercise, we found the derivative of \(y = \sin(x^2 + x)\), arriving at a formula that represents every potential point of instantaneous change within this given function. For every value of \(x\), we can predict how \(y\) will shift.

• Finding a derivative gives you a roadmap for how a curve behaves.
• It helps in understanding increasing, decreasing, and critical points of the function.

Mastering derivatives unlocks deeper insights into mathematical relationships and predictions across multiple domains.

Mathematics Education

Learning calculus, including the process of differentiation, serves as a stepping stone for students venturing into higher learning and analytical thinking. It nurtures critical problem-solving skills by continually challenging students to break down problems and apply rules like the chain rule.

Mathematics education often involves exploring diverse functions and their derivatives. Concepts like trigonometric derivatives, composite functions, and application of rules such as the chain rule enable learners to grasp more complicated situations.

• Every new concept builds upon previous knowledge, allowing deeper understanding.
• Mastering differentiation opens doors to advanced fields like physics and engineering.

Through consistent practice and curiosity, students can transform calculus from a daunting subject to a powerful tool in their educational toolkit.