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Chapter 3: Problem 13

\(7 - 46\) Find the derivative of the function. $$y = \cos \left( a ^ { 3 } + x ^ { 3 } \right)$$

Short Answer

Expert verified
The derivative is \(-3x^2 \sin(a^3 + x^3)\).

Step by step solution

01

Identify and Apply the Chain Rule

To differentiate the function \(y = \cos(a^3 + x^3)\), we recognize that it is a composite function. The outer function is \(\cos(u)\) and the inner function is \(u = a^3 + x^3\). We'll apply the chain rule, which states that the derivative of \(y\) with respect to \(x\) is the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function with respect to \(x\).
02

Differentiate the Outer Function

The derivative of the outer function \(\cos(u)\) with respect to \(u\) is \(-\sin(u)\). So the first part of the derivative is \(-\sin(a^3 + x^3)\).
03

Differentiate the Inner Function

The inner function is \(u = a^3 + x^3\). Differentiating \(a^3 + x^3\) with respect to \(x\), treating \(a\) as a constant, gives \(3x^2\). Thus, the derivative of the inner function with respect to \(x\) is \(3x^2\).
04

Combine Using the Chain Rule

By the chain rule, the derivative \(\frac{dy}{dx}\) is obtained by multiplying the derivatives of the outer and inner functions. Combine them to get: \(\frac{dy}{dx} = -\sin(a^3 + x^3) \times 3x^2\). This simplifies to \(\frac{dy}{dx} = -3x^2 \sin(a^3 + x^3)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative
A derivative represents the rate at which a function changes as its input changes. It's like measuring the speed of change. In our exercise, the derivative helps us find how fast the value of the cosine function is changing with respect to changes in the variable, specifically the variable \( x \). In simpler terms, it tells us the "slope" of the function at any given point, providing insight into the function's behavior and trajectory. Derivatives are foundational to calculus. They help solve problems involving motion, growth, decay, and optimization in various fields such as physics, economics, and engineering.
Composite Function
A composite function is simply a function within another function. In our example, we look at \( y = \cos(a^3 + x^3) \). Here, \( y \) is expressed as the cosine of some inner expression, \( a^3 + x^3 \). Therefore, \( \cos(u) \) is the outer function, and \( u = a^3 + x^3 \) is the inner function. Understanding composite functions is critical when applying the chain rule, as the chain rule allows for the differentiation of these multi-layered functions by differentiating each layer step by step. This approach simplifies finding the derivative of complex functions.
Trigonometric Function
Trigonometric functions, like sine and cosine, describe relationships in triangles and are fundamental in modeling periodic phenomena like waves and oscillations. In our exercise, \( \cos \) is the trigonometric function we're dealing with. These functions have unique derivatives: the derivative of \( \cos(u) \) is \( -\sin(u) \). We use this fact directly during differentiation. Trigonometric functions appear frequently in calculus, and their derivatives often need to be applied using rules like the chain rule to integrate seamlessly into problems involving composite or complicated functions.
Differentiation
Differentiation is the process of finding the derivative of a function and is a critical operation in calculus. It provides a means to determine how a function's output value changes with respect to changes in its input value. In our problem, we use differentiation to determine the rate of change in the function \( y = \cos(a^3 + x^3) \) relative to the variable \( x \). Differentiation involves various rules and techniques, including the chain rule, product rule, and quotient rule, helping solve a wide array of problems by breaking down complex expressions into manageable parts allowing more straightforward computation of derivatives.

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