Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} e^{-1 / x^{2}} $$

Step-by-Step Solution

Answer

The derivative is \( 2x^{-3} e^{-1/x^2} \).

Step 1: Identify the Derivative Rule Needed

The function given is \( e^{-1/x^2} \). Notice that it's an exponential function where the exponent itself is a function of \( x \). This means we'll need to use the chain rule for differentiation.

Step 2: Differentiate the Outer Function

The outer function is \( e^u \) where \( u = -\frac{1}{x^2} \). The derivative of \( e^u \) with respect to \( u \) is simply \( e^u \). So, we leave \( e^{-1/x^2} \) as it is for now.

Step 3: Differentiate the Inner Function

The inner function is \( u = -\frac{1}{x^2} \). To differentiate this, write it as \( u = -(x^{-2}) \). The derivative of \( x^{-2} \) is \( -2x^{-3} \). Thus, differentiating \( u \) we get \( \frac{du}{dx} = 2x^{-3} \).

Step 4: Apply the Chain Rule

The chain rule states that \( \frac{d}{dx} e^{u} = e^u \cdot \frac{du}{dx} \). Substitute \( e^{-1/x^2} \) for \( e^u \) and \( 2x^{-3} \) for \( \frac{du}{dx} \).

Step 5: Simplify

Now, multiply: \( D_x e^{-1/x^2} = e^{-1/x^2} \cdot 2x^{-3} = 2x^{-3} e^{-1/x^2} \). This is the derivative of the function.

Key Concepts

Understanding the Chain Rule

The chain rule is a fundamental concept in calculus, specifically in differentiation. It is used when differentiating a composition of functions, which is a function nested within another function.
Imagine it like peeling layers of an onion: you have to work through each layer one at a time.
In simpler terms, the chain rule states that to differentiate a composite function, you first differentiate the outer function and then multiply that by the derivative of the inner function. This approach is very powerful when dealing with complex functions.

• First identify the outer and inner functions.
• Differentiate the outer function.
• Find the derivative of the inner function.
• Multiply these derivatives together.

The notation for the chain rule can be expressed as:\[\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)\]It ensures you keep track of each part of a composite function when finding its derivative.

What is an Exponential Function?

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. In our exercise, the function is \( e^{-1/x^2} \), where \( e \) is the base and \(-\frac{1}{x^2}\) is the exponent.
Exponential functions are prevalent in natural processes like population growth and radioactive decay, owing to their unique properties.Some key features of exponential functions include:

• The base is a constant, often Euler's number \( e \approx 2.718 \).
• The exponent is a variable expression.
• They grow or decay at a rate proportional to their current value.

In differentiation, the exponential function \( e^u \) has a unique property: its derivative with respect to \( u \) is the function itself, \( e^u \). This property makes handling exponential functions quite straightforward in calculus.

Differentiating the Inner Function

The inner function in the exercise is \( u = -\frac{1}{x^2} \). It is key, because the chain rule requires us to differentiate this part separately before combining it with the outer function differentiation.
To differentiate \( u \), we first rewrite it in a form that's easier to handle: \( u = -(x^{-2}) \). This transformation makes differentiation more straightforward as the power rule can be directly applied.
Let's apply the power rule: the derivative of \( x^n \) is \( nx^{n-1} \).
Hence, \( \frac{d}{dx} x^{-2} = -2x^{-3} \). Since our function is \( u = -(x^{-2}) \), the derivative is \(-(-2x^{-3}) = 2x^{-3} \).
Understanding how to break down and differentiate the inner function allows us to use the chain rule effectively, ensuring that all parts of the composite function are accounted for in the final derivative.