Problem 11 - Applied Calculus

Question

Differentiate the functions in Problems 1-20. Assume that \(A\) and \(B\) are constants. $$ y=\sin \left(x^{2}\right) $$

Step-by-Step Solution

Answer

The derivative is \(y' = 2x \cos(x^2)\).

Step 1: Identify the Differentiation Rule

The function given is a composite function because it involves a trigonometric function and a power of x. To differentiate this function, we will need to apply the chain rule. The chain rule states that if you have a composite function \(y = f(g(x))\), then its derivative is given by \(y' = f'(g(x)) \cdot g'(x)\).

Step 2: Differentiate the Outer Function

For the function \(y = \sin(x^2)\), consider \(f(u) = \sin(u)\) where \(u = x^2\). The derivative of \(f(u) = \sin(u)\) is \(f'(u) = \cos(u)\).

Step 3: Differentiate the Inner Function

Next, differentiate the inner function \(u = x^2\). The derivative of \(u = x^2\) is \(u' = 2x\).

Step 4: Apply the Chain Rule

Use the chain rule to find the derivative of the composite function \(y = \sin(x^2)\). Substitute the results from Steps 2 and 3 into the chain rule: \[ y' = f'(u) \cdot u' = \cos(x^2) \cdot (2x) \].

Step 5: Simplify the Result

The derivative of \(y = \sin(x^2)\) is simplified to: \[ y' = 2x \cos(x^2) \].

Key Concepts

Chain Rule

The chain rule is a powerful tool in calculus used to differentiate composite functions. It allows us to find the derivative of a function that is made up of two or more functions. Imagine functions stacked within each other, like Russian dolls. For example, if you have a function inside another function, such as \( f(g(x)) \), the chain rule helps to "unwrap" the layers.
To apply the chain rule, you follow these steps:

• First, identify the outer function \( f \) and the inner function \( g \).
• Differentiate the outer function, treating the inner function as if it were a simple variable.
• Differentiate the inner function as if the outer function isn't there.
• Multiply these derivatives together: \( y' = f'(g(x)) \cdot g'(x) \).

In our example, with \( y = \sin(x^2) \), we recognize \( \sin(u) \) as the outer function where \( u = x^2 \). The derivative of \( \sin(u) \) is \( \cos(u) \), and the derivative of \( u = x^2 \) is \( 2x \). Using the chain rule, we find that the derivative is \( 2x \cos(x^2) \). This approach simplifies and clarifies the process of differentiating complex functions.

Trigonometric Functions

Trigonometric functions are a key part of calculus, appearing often in both practical and theoretical problems. These functions include \( \sin, \cos, \) and \( \tan \), each with unique characteristics and derivatives.
For a function like \( \sin(x) \), the derivative is straightforward: \( \cos(x) \). Similarly, for \( \cos(x) \), the derivative is \( -\sin(x) \). Knowing these basic derivatives is crucial for solving more complex problems involving trigonometric functions.
In the context of our exercise, \( y = \sin(x^2) \) involves \( \sin \), but since it's a composite function, we employ the chain rule. First, we differentiate \( \sin \) with respect to the inner function \( x^2 \), giving us \( \cos(x^2) \). Because all trigonometric functions are periodic, they provide unique properties such as repeating values, which can also influence how we approach calculus problems.

Composite Functions

Composite functions are functions made up by combining two or more functions. They take on the form \( f(g(x)) \) where one function is applied to the results of another function. This layer-like structure makes them slightly more complex to differentiate compared to simple functions.
However, the good news is that the chain rule can effectively handle composite functions. When you break down \( y = \sin(x^2) \), \( \sin \) is the outer function and \( x^2 \) is the inner function. Each part needs to be differentiated separately according to their respective rules before applying the chain rule to combine the results.
Understanding composite functions is essential because they represent a large number of real-world situations. For instance, when studying physical phenomena or in engineering, systems are often described by composite relationships rather than simple ones. Therefore, mastering them through the chain rule becomes a vital calculus skill.