Problem 27 - Thomas Calculus

Question

In Exercises \(17-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\cos \left(e^{-\theta^{2}}\right) $$

Step-by-Step Solution

Answer

The derivative is \( 2\theta e^{-\theta^2} \sin(e^{-\theta^2}) \).

Step 1: Identify the Function Structure

The given function is a composite function, where the outer function is the cosine function, and the inner function is the exponential function raised to the negative square of \( \theta \). This structure requires the use of the chain rule for differentiation.

Step 2: Apply the Chain Rule

The chain rule states that if \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \). Here, \( f(u) = \cos(u) \) and \( u = e^{-\theta^2} \). We need to differentiate the cosine function with respect to \( u \) and multiply by the derivative of the inner function \( u \) with respect to \( \theta \).

Step 3: Differentiate the Outer Function

Differentiate \( f(u) = \cos(u) \) with respect to \( u \). The derivative provided by the derivative identity is \( f'(u) = -\sin(u) \). So, \( f'(e^{-\theta^2}) = -\sin(e^{-\theta^2}) \).

Step 4: Differentiate the Inner Function

For the inner function \( u = e^{-\theta^2} \), apply the chain rule: \( \frac{du}{d\theta} = e^{-\theta^2} \cdot \frac{d}{d\theta}(-\theta^2) \). Calculate \( \frac{d}{d\theta}(-\theta^2) = -2\theta \), thus \( \frac{du}{d\theta} = e^{-\theta^2} \cdot (-2\theta) = -2\theta e^{-\theta^2} \).

Step 5: Combine the Results

Combine the derivatives from Steps 3 and 4. Multiply the derivative of the outer function by the derivative of the inner function: \( \frac{dy}{d\theta} = -\sin(e^{-\theta^2}) \cdot (-2\theta e^{-\theta^2}) \).

Step 6: Simplify the Expression

Simplify the expression obtained in Step 5. The product becomes: \( \frac{dy}{d\theta} = 2\theta e^{-\theta^2} \sin(e^{-\theta^2}) \). This is the final expression for the derivative of \( y \) with respect to \( \theta \).

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. It is crucial when you have functions nested within each other. This rule states that if you have a function that can be expressed as the composition of two functions, say \( y = f(g(x)) \), the derivative of \( y \) with respect to \( x \) is obtained by multiplying the derivative of the outer function by the derivative of the inner function.
Here's a simple way to understand it:

• Identify the outer function, \( f \), and find its derivative.
• Identify the inner function, \( g \), and find its derivative.
• Multiply these derivatives as per the rule: \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

In our exercise, identifying \( \cos(u) \) as the outer function, and \( e^{-\theta^2} \) as the inner function, is the key step. Applying the chain rule, you differentiate the outer function with respect to the inner one, then multiply by the derivative of the inner function respecting its variable.

Composite Function

A composite function is formed when one function is applied to the result of another. This means that the output of the inner function becomes the input of the outer function.
In our example, the composite nature is seen in:

• Outer function: the cosine function \( \cos(u) \).
• Inner function: the exponential function \( e^{-\theta^2} \).

Understanding this structure is fundamental before attempting any differentiation. Recognizing a composite function allows you to apply strategies like the chain rule effectively. You always want to break the function down into its components, so you can tackle each separately before combining the results.

Differentiation

Differentiation is the process used to find the derivative of a function, essentially giving us the rate at which a function is changing at any given point. This is a cornerstone of calculus and is especially powerful when working with more complex functions involving multiple layers or compositions.
The chain rule is a vital tool in this process, specifically for functions involving layers upon layers like composites. During differentiation:

• You need to recognize any composite functions, as these will require the chain rule.
• Proceed by finding derivatives layer by layer, starting with the innermost to the outermost.
• Combine these derivatives as needed to construct the final derivative expression.

In our step-by-step solution, the differentiation started with simplifying the initial composite structure. Each derivative was calculated step by step until the complete expression was obtained.