Problem 51 - Calculus Early Transcendentals

Question

Compute the following derivatives using the method of your choice. $$\frac{d}{d x}\left(e^{-10 x^{2}}\right)$$

Step-by-Step Solution

Answer

Answer: The derivative of the function $$y=e^{-10x^2}$$ is $$\frac{dy}{dx} = 20x \cdot e^{-10x^2}$$

Step 1: Identify the inner function

Let's identify the inner function. In this case, it is $$u = -10x^₂$$.

Step 2: Compute the derivative of the inner function

Now we need to compute the derivative of the inner function, $$u = -10x^2$$, with respect to x. Using the power rule, we have: $$\frac{du}{dx} = \frac{d}{dx} (-10x^2) = -20x$$

Step 3: Compute the derivative of the outer function

We want to compute the derivative of the outer function, $$e^{-u}$$, with respect to u. Using the derivative properties of the exponential function, we have: $$\frac{d}{d u}\left[e^{-u}\right] = -e^{-u}$$

Step 4: Apply the chain rule

Now we will apply the chain rule to compute the derivative of the given function. Replace u with -10x^2 and substitute the values we found in steps 2 and 3: $$\frac{d}{d x}\left[e^{-10 x^{2}}\right] = -e^{-(-10x^2)} \cdot (-20x)$$

Step 5: Simplify the expression

Multiply the scalar values to simplify the expression: $$\frac{d}{d x}\left[e^{-10 x^{2}}\right] = 20x \cdot e^{-10x^2}$$ So, the derivative of the given function is: $$\frac{d}{d x}\left(e^{-10 x^{2}}\right) = 20x \cdot e^{-10x^2}$$

Key Concepts

Understanding the Chain Rule

The chain rule is a fundamental tool in calculus, especially useful when dealing with composite functions. In simple terms, the chain rule helps us find the derivative of a function that is composed of two or more other functions.
To apply the chain rule, we:

• Identify the inner function and the outer function.
• Take the derivative of the outer function, while leaving the inner function unchanged.
• Multiply the result by the derivative of the inner function.

This method is crucial when dealing with more complex expressions where one function is nested inside another, like in the problem of finding the derivative of \(e^{-10x^2}\). By using the chain rule, we handle each function separately and combine them for the final derivative.

Mastering the Power Rule

The power rule is one of the most basic yet powerful rules in calculus for finding derivatives. It states that for any function of the form \(x^n\), where \(n\) is a constant, the derivative is \(nx^{n-1}\).
Using the power rule simplifies the process of finding derivatives, making it a go-to rule for polynomial functions. For instance, in the expression \(-10x^2\) from our exercise, applying the power rule:

• The exponent is \(2\).
• We multiply by the coefficient and decrease the exponent by one, resulting in \(-20x\).

Understanding the power rule in depth ensures we can handle any polynomial term quickly and efficiently. It cuts down on computation time and complexity, leaving us with the right derivative expression.

Exponential Function Derivatives

The exponential function, often denoted as \(e^x\), has a distinct derivative property that makes it unique. The key point about the derivative of \(e^x\) is that it remains unchanged, meaning \(\frac{d}{dx}(e^x) = e^x\). However, things change slightly when we deal with \(e^u\) where \(u\) is a function of \(x\).
When differentiating \(e^u\), we:

• Find the derivative of the outer \(e^u\), which is \(e^u\) itself.
• Multiply by the derivative of the inner function \(u\).

For example, given \(e^{-10x^2}\), the outer function's derivative is \(-e^{-u}\), where \(-u=-10x^2\). We then multiply by the derivative of \(-10x^2\).

This is where our understanding of both exponential functions and the chain rule comes into play, allowing us to effectively break down and solve the derivative step-by-step.