Calculating the First, Second, and Third Derivatives of f(x) e^(-x^2/2)

In this article, we will explore the process of calculating the first, second, and third derivatives of the function f(x) e^(-x^2/2). This is a common example in calculus, particularly when dealing with exponential functions and understanding their rates of change.

1. The First Derivative

We start by finding the first derivative of the function. The derivative of an exponential function can often be found using the chain rule. For the given function:

f(x) e^(-x^2/2)

Applying the chain rule, d(e^u)/dx e^u . u', where u -x^2/2, we get:

f'(x) -x e^(-x^2/2)

The product rule is also involved here. Given g(x) e^(-x^2/2) and h(x) -x, we can write:

g'(x)h(x) g(x)h'(x) -x e^(-x^2/2) e^(-x^2/2) * (-1) -x e^(-x^2/2) - e^(-x^2/2)

Upon simplification, we get the first derivative as:

f'(x) -x e^(-x^2/2)

2. The Second Derivative

Next, let's find the second derivative of the function f(x). We differentiate the first derivative with respect to x. Applying the product rule again:

u -x, v e^(-x^2/2) > u' -1, v' e^(-x^2/2) * (-2x)

The second derivative is then:

f''(x) u'v uv' (-1) * e^(-x^2/2) (-x) * (e^(-x^2/2) * (-2x))

Simplifying this expression, we get:

f''(x) -e^(-x^2/2) 2x^2 e^(-x^2/2) (2x^2 - 1) e^(-x^2/2)

3. The Third Derivative

Lastly, we will calculate the third derivative. We differentiate the second derivative with respect to x once more:

We use the product rule similarly:

u 2x^2 - 1, v e^(-x^2/2) > u' 4x, v' e^(-x^2/2) * (-2x)

The third derivative is:

f'''(x) u'v uv' (4x) * e^(-x^2/2) (2x^2 - 1) * (e^(-x^2/2) * (-2x))

After simplifying, we get:

f'''(x) 4x e^(-x^2/2) - 2x(2x^2 - 1) e^(-x^2/2) 4x e^(-x^2/2) - 4x^3 e^(-x^2/2) 2x e^(-x^2/2)

Combining like terms, the result is:

f'''(x) -4x^3 e^(-x^2/2) 6x e^(-x^2/2)

Conclusion

Understanding the first, second, and third derivatives of the function f(x) e^(-x^2/2) helps us gain insights into the behavior of the function at any point. The process involves the chain rule and product rule, which are fundamental concepts in calculus. These derivatives are particularly useful in applications such as optimization, physics, and data science.

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Further Reading

If you're interested in exploring more calculus concepts, you might want to check out resources on optimization techniques, different identities in calculus, or applications of derivatives in real-world scenarios.