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Chapter 3: Problem 18

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=e^{-x^{2}}$$

Short Answer

Expert verified
Question: Find the derivative of the function \(y = e^{-x^2}\) with respect to \(x\). Answer: The derivative of the function \(y = e^{-x^2}\) with respect to \(x\) is \(\frac{dy}{dx} = -2xe^{-x^2}\).

Step by step solution

01

Identify the inner and outer functions

In this case, we have \(y = e^{-x^2}\). The inner function is \(g(x) = -x^2\) and the outer function is \(f(u) = e^u\).
02

Find the derivative of the outer function

To compute the derivative of \(f(u) = e^u\) with respect to \(u\), we can use the fact that the derivative of \(e^u\) is simply \(e^u\). So, we have: $$f'(u) = e^u$$
03

Find the derivative of the inner function

To compute the derivative of \(g(x) = -x^2\) with respect to \(x\), we can use the power rule. The power rule states that if \(g(x) = x^n\), then \(g'(x) = nx^{n-1}\). So, in our case: $$g'(x) = -2x$$
04

Apply the Chain Rule

To find the derivative of \(y = f(g(x))\), we can now apply the Chain Rule: $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$ Substitute the derivatives we found in Steps 2 and 3: $$\frac{dy}{dx} = e^{-x^2} \cdot(-2x)$$ Therefore, the derivative of \(y = e^{-x^2}\) with respect to \(x\) is: $$\frac{dy}{dx} = -2xe^{-x^2}$$

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