If $f(x) = 3 \arctan \left(8 E^x\right$\], Find $f^{\prime}(x$\].

Feb 26, 2025 by ADMIN 66 views

If $f(x) = 3 \arctan \left(8 e^x\right)$ , find $f^{\prime}(x)$

In this article, we will explore the concept of finding the derivative of a function that involves the arctan function and exponential function. The arctan function is the inverse of the tangent function, and it is used to find the angle whose tangent is a given number. The exponential function is a fundamental function in mathematics that is used to model population growth, chemical reactions, and other phenomena.

The function we are given is $f(x) = 3 \arctan \left(8 e^x\right)$ . This function involves the arctan function and the exponential function. The arctan function takes the input $8 e^x$ and returns the angle whose tangent is $8 e^x$ . The exponential function $e^x$ is a fundamental function that is used to model population growth, chemical reactions, and other phenomena.

To find the derivative of the function $f(x) = 3 \arctan \left(8 e^x\right)$ , we will use the chain rule and the fact that the derivative of the arctan function is $\frac{1}{1 + x^2}$ . 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 $f^{\prime}(g(x)) \cdot g^{\prime}(x)$ .

Step 1: Find the Derivative of the Arctan Function

The derivative of the arctan function is given by $\frac{1}{1 + x^2}$ . This is a fundamental result in calculus that can be proven using the definition of the derivative.

Step 2: Find the Derivative of the Exponential Function

The derivative of the exponential function $e^x$ is given by $e^x$ . This is a fundamental result in calculus that can be proven using the definition of the derivative.

Step 3: Apply the Chain Rule

Using the chain rule, we can find the derivative of the composite function $f(x) = 3 \arctan \left(8 e^x\right)$ . The derivative of the composite function is given by:

f^{\prime}(x) = 3 \cdot \frac{1}{1 + (8 e^x)^2} \cdot 8 e^x

Simplifying the Derivative

We can simplify the derivative by combining the constants and the exponential function:

f^{\prime}(x) = \frac{24 e^x}{1 + 64 e^{2x}}

In this article, we have found the derivative of the function $f(x) = 3 \arctan \left(8 e^x\right)$ using the chain rule and the fact that the derivative of the arctan function is $\frac{1}{1 + x^2}$ . The derivative of the composite function is given by:

f^{\prime}(x) = \frac{24 e^x}{1 + 64 e^{2x}}

This result can be used to model population growth, chemical reactions, and other phenomena that involve the arctan function and the exponential function.

For further reading on the topic of derivatives and the chain rule, we recommend the following resources:

Arctan Function: The inverse of the tangent function, which returns the angle whose tangent is a given number.
Exponential Function: A fundamental function that is used to model population growth, chemical reactions, and other phenomena.
Chain Rule: A fundamental result in calculus that states that if we have a composite function of the form $f(g(x))$ , then the derivative of the composite function is given by $f^{\prime}(g(x)) \cdot g^{\prime}(x)$ .
Q&A: If $f(x) = 3 \arctan \left(8 e^x\right)$ , find $f^{\prime}(x)$ ===========================================================

In our previous article, we found the derivative of the function $f(x) = 3 \arctan \left(8 e^x\right)$ using the chain rule and the fact that the derivative of the arctan function is $\frac{1}{1 + x^2}$ . In this article, we will answer some common questions related to the derivative of the function $f(x) = 3 \arctan \left(8 e^x\right)$ .

Q: What is the derivative of the arctan function?

A: The derivative of the arctan function is given by $\frac{1}{1 + x^2}$ . This is a fundamental result in calculus that can be proven using the definition of the derivative.

Q: What is the derivative of the exponential function?

A: The derivative of the exponential function $e^x$ is given by $e^x$ . This is a fundamental result in calculus that can be proven using the definition of the derivative.

Q: How do I apply the chain rule to find the derivative of the composite function?

A: To apply the chain rule, you need to find the derivative of the outer function and the derivative of the inner function. Then, you multiply the two derivatives together to get the derivative of the composite function.

Q: What is the derivative of the composite function $f(x) = 3 \arctan \left(8 e^x\right)$ ?

A: The derivative of the composite function $f(x) = 3 \arctan \left(8 e^x\right)$ is given by:

f^{\prime}(x) = \frac{24 e^x}{1 + 64 e^{2x}}

Q: Can I use the chain rule to find the derivative of any composite function?

A: Yes, you can use the chain rule to find the derivative of any composite function. 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 $f^{\prime}(g(x)) \cdot g^{\prime}(x)$ .

Q: What are some common applications of the chain rule?

A: The chain rule has many applications in calculus, including:

Finding the derivative of composite functions
Finding the derivative of functions that involve trigonometric functions
Finding the derivative of functions that involve exponential functions
Finding the derivative of functions that involve logarithmic functions

Q: Can I use the chain rule to find the derivative of a function that involves multiple composite functions?

A: Yes, you can use the chain rule to find the derivative of a function that involves multiple composite functions. The chain rule can be applied multiple times to find the derivative of a function that involves multiple composite functions.

In this article, we have answered some common questions related to the derivative of the function $f(x) = 3 \arctan \left(8 e^x\right)$ . We have also discussed the chain rule and its applications in calculus. We hope that this article has been helpful in understanding the derivative of the function $f(x) = 3 \arctan \left(8 e^x\right)$ .