Computing the First Derivative

Oct 13, 2025 by windailys.com 88 views

Introduction

In calculus, the first derivative of a function represents the rate of change of the function with respect to its input. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore the process of computing the first derivative of a function, using the example of the function $f(x) = \exp(-x^2)$ .

The Function

The function $f(x) = \exp(-x^2)$ is a continuous and differentiable function that maps real numbers to real numbers. The function is defined as the exponential of the negative square of the input variable $x$ . This function is often used in mathematical modeling to describe phenomena that exhibit exponential decay.

Computing the First Derivative

To compute the first derivative of the function $f(x) = \exp(-x^2)$ , we will use the chain rule and the fact that the derivative of the exponential function is the exponential function itself.

Step 1: Apply the Chain Rule

The chain rule states that if we have a composite function of the form $f(g(x))$ , then the derivative of the composite function is given by $f'(g(x)) \cdot g'(x)$ . In this case, we have $f(x) = \exp(-x^2)$ , which can be written as $f(g(x))$ , where $g(x) = -x^2$ .

Step 2: Compute the Derivative of the Inner Function

The derivative of the inner function $g(x) = -x^2$ is given by $g'(x) = -2x$ .

Step 3: Compute the Derivative of the Outer Function

The derivative of the outer function $f(x) = \exp(x)$ is given by $f'(x) = \exp(x)$ .

Step 4: Apply the Chain Rule

Now that we have computed the derivatives of the inner and outer functions, we can apply the chain rule to compute the derivative of the composite function.

\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) = \exp(-x^2) \cdot (-2x) = -2x \exp(-x^2)

Conclusion

In this article, we have computed the first derivative of the function $f(x) = \exp(-x^2)$ using the chain rule. The derivative of the function is given by $-2x \exp(-x^2)$ . This result can be used to analyze the behavior of the function and to solve optimization problems.

Real Numbers with Positive Values

Let $a, b, c$ be real numbers with $a, b > 0$ . We can use the result from the previous section to compute the derivative of the function $f(x) = \exp(-x^2)$ .

Step 1: Substitute the Values of $a$ and $b$

We can substitute the values of $a$ and $b$ into the function $f(x) = \exp(-x^2)$ to obtain $f(x) = \exp(-ax^2)$ .

Step 2: Compute the Derivative

Using the result from the previous section, we can compute the derivative of the function $f(x) = \exp(-ax^2)$ .

\frac{d}{dx} f(x) = -2ax \exp(-ax^2)

Optimization Problems

The derivative of the function $f(x) = \exp(-ax^2)$ can be used to solve optimization problems. For example, we can use the derivative to find the maximum or minimum value of the function.

Step 1: Find the Critical Points

To find the maximum or minimum value of the function, we need to find the critical points. The critical points are the values of $x$ that make the derivative equal to zero.

-2ax \exp(-ax^2) = 0

Step 2: Solve for $x$

We can solve for $x$ by setting the exponent equal to zero.

-ax^2 = 0

Step 3: Find the Maximum or Minimum Value

Once we have found the critical points, we can use the second derivative test to determine whether the critical point corresponds to a maximum or minimum value.

Conclusion

In this article, we have computed the first derivative of the function $f(x) = \exp(-x^2)$ using the chain rule. We have also used the result to solve optimization problems. The derivative of the function can be used to analyze the behavior of the function and to solve optimization problems.

Real-World Applications

The derivative of the function $f(x) = \exp(-x^2)$ has numerous real-world applications. For example, the function can be used to model the behavior of a population that exhibits exponential decay.

Step 1: Model the Population

We can use the function $f(x) = \exp(-ax^2)$ to model the population.

Step 2: Compute the Derivative

Using the result from the previous section, we can compute the derivative of the function $f(x) = \exp(-ax^2)$ .

\frac{d}{dx} f(x) = -2ax \exp(-ax^2)

Step 3: Analyze the Behavior

We can use the derivative to analyze the behavior of the population.

Conclusion

$In this article, we have computed the first derivative of the function $f(x) = \exp(-x^2)$ using the chain rule. We have also used the result to solve optimization problems and to model the behavior of a population that exhibits exponential decay. The derivative of the function can be used to analyze the behavior of the function and to solve optimization problems.<br/> **Q&A: Computing the First Derivative of a Function**$

Q: What is the first derivative of a function?

A: The first derivative of a function represents the rate of change of the function with respect to its input. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics.

Q: How do I compute the first derivative of a function?

A: To compute the first derivative of a function, you can use the following steps:

Identify the function and its input variable.
Apply the chain rule and the fact that the derivative of the exponential function is the exponential function itself.
Compute the derivative of the inner function.
Compute the derivative of the outer function.
Apply the chain rule to compute the derivative of the composite function.

Q: What is the chain rule?

A: The chain rule is a mathematical concept that allows us to compute the derivative of a composite function. It states that if we have a composite function of the form $f(g(x))$ , then the derivative of the composite function is given by $f'(g(x)) \cdot g'(x)$ .

Q: How do I apply the chain rule?

A: To apply the chain rule, you need to follow these steps:

Identify the inner function $g(x)$ and its derivative $g'(x)$ .
Identify the outer function $f(x)$ and its derivative $f'(x)$ .
Compute the derivative of the composite function using the formula $f'(g(x)) \cdot g'(x)$ .

Q: What is the derivative of the exponential function?

A: The derivative of the exponential function is the exponential function itself. In other words, if we have a function of the form $f(x) = \exp(x)$ , then the derivative of the function is given by $f'(x) = \exp(x)$ .

Q: How do I compute the derivative of a composite function?

A: To compute the derivative of a composite function, you can use the following steps:

Identify the inner function $g(x)$ and its derivative $g'(x)$ .
Identify the outer function $f(x)$ and its derivative $f'(x)$ .
Compute the derivative of the composite function using the formula $f'(g(x)) \cdot g'(x)$ .

Q: What is the significance of the first derivative of a function?

Q: How do I use the first derivative of a function to solve optimization problems?

A: To use the first derivative of a function to solve optimization problems, you can follow these steps:

Compute the first derivative of the function.
Find the critical points by setting the derivative equal to zero.
Use the second derivative test to determine whether the critical point corresponds to a maximum or minimum value.

Q: What is the second derivative test?

A: The second derivative test is a mathematical concept that allows us to determine whether a critical point corresponds to a maximum or minimum value. It states that if we have a function $f(x)$ and a critical point $x = c$ , then if the second derivative of the function at the critical point is positive, the critical point corresponds to a minimum value. If the second derivative of the function at the critical point is negative, the critical point corresponds to a maximum value.

Q: How do I use the second derivative test?

A: To use the second derivative test, you need to follow these steps:

Compute the first derivative of the function.
Find the critical points by setting the derivative equal to zero.
Compute the second derivative of the function at the critical point.
Use the second derivative test to determine whether the critical point corresponds to a maximum or minimum value.

Conclusion

In this article, we have answered some of the most frequently asked questions about computing the first derivative of a function. We have covered topics such as the chain rule, the derivative of the exponential function, and the second derivative test. We hope that this article has been helpful in providing you with a better understanding of the first derivative of a function and its applications.