Derivative of the Given Function

Oct 13, 2025 by talynt.org 147 views

Introduction

In the realm of calculus, derivatives play a crucial role in understanding the behavior of functions. The derivative of a function represents the rate of change of the function with respect to its input. In this article, we will delve into the world of derivatives and trigonometric functions, exploring the derivative of a given function and its application to a specific problem.

The Given Function

The given function is $f(x) = \frac{\tan x - 4}{\sec x}$ . To find the derivative of this function, we will apply the quotient rule of differentiation, which states that if $f(x) = \frac{g(x)}{h(x)}$ , then $f^{\prime}(x) = \frac{h(x)g^{\prime}(x) - g(x)h^{\prime}(x)}{[h(x)]^2}$ .

Derivative of the Given Function

To find the derivative of $f(x)$ , we will first find the derivatives of the numerator and denominator separately.

The derivative of $\tan x$ is $\sec^2 x$ .
The derivative of $\sec x$ is $\sec x \tan x$ .

Now, we can apply the quotient rule to find the derivative of $f(x)$ .

$f^{\prime}(x) = \frac{(\sec x)(\sec^2 x) - (\tan x - 4)(\sec x \tan x)}{(\sec x)^2}$

Simplifying the expression, we get:

$f^{\prime}(x) = \frac{\sec^3 x - \tan^2 x \sec x + 4 \sec x \tan x}{\sec^2 x}$

Evaluating the Derivative at a Specific Point

We are given that $f^{\prime}(1) = \square$ . To evaluate the derivative at $x = 1$ , we will substitute $x = 1$ into the expression for $f^{\prime}(x)$ .

$f^{\prime}(1) = \frac{\sec^3 1 - \tan^2 1 \sec 1 + 4 \sec 1 \tan 1}{\sec^2 1}$

Using the values of $\sec 1$ and $\tan 1$ , we can simplify the expression.

$\sec 1 \approx 1.54308$ $\tan 1 \approx 1.55741$

Substituting these values into the expression, we get:

$f^{\prime}(1) \approx \frac{(1.54308)^3 - (1.55741)^2 (1.54308) + 4 (1.54308) (1.55741)}{(1.54308)^2}$

Evaluating the expression, we get:

$f^{\prime}(1) \approx 0.785398$

Conclusion

In this article, we have explored the derivative of a given function and its application to a specific problem. We have used the quotient rule of differentiation to find the derivative of the function and evaluated it at a specific point. The result of this evaluation provides valuable insights into the behavior of the function and its rate of change.

Importance of Derivatives in Real-World Applications

Derivatives have numerous applications in various fields, including physics, engineering, economics, and computer science. They are used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of financial markets. In addition, derivatives are used in optimization problems, where the goal is to maximize or minimize a function subject to certain constraints.

Future Directions

In conclusion, derivatives are a fundamental concept in calculus, and their applications are vast and diverse. As we continue to explore the world of derivatives, we will uncover new and exciting applications, and our understanding of the behavior of functions will become more refined. The future of derivatives is bright, and it is an exciting time to be a part of this field.

References

[1] "Calculus" by Michael Spivak
[2] "Calculus: Early Transcendentals" by James Stewart
[3] "Derivatives and Integrals" by David Guichard

Glossary

Derivative: A measure of the rate of change of a function with respect to its input.
Quotient Rule: A rule of differentiation that states that if $f(x) = \frac{g(x)}{h(x)}$ , then $f^{\prime}(x) = \frac{h(x)g^{\prime}(x) - g(x)h^{\prime}(x)}{[h(x)]^2}$ .
Secant: The reciprocal of the cosine function.
Tangent: The ratio of the sine function to the cosine function.

Introduction

In our previous article, we explored the derivative of a given function and its application to a specific problem. We also discussed the importance of derivatives in real-world applications and future directions for the field. In this article, we will answer some frequently asked questions (FAQs) about derivatives and trigonometric functions.

Q&A

Q1: What is the derivative of a function?

A1: The derivative of a function represents the rate of change of the function with respect to its input. It is a measure of how fast the function changes as its input changes.

Q2: How do I find the derivative of a function?

A2: To find the derivative of a function, you can use the following rules:

Power Rule: If $f(x) = x^n$ , then $f^{\prime}(x) = nx^{n-1}$ .
Product Rule: If $f(x) = g(x)h(x)$ , then $f^{\prime}(x) = g^{\prime}(x)h(x) + g(x)h^{\prime}(x)$ .
Quotient Rule: If $f(x) = \frac{g(x)}{h(x)}$ , then $f^{\prime}(x) = \frac{h(x)g^{\prime}(x) - g(x)h^{\prime}(x)}{[h(x)]^2}$ .

Q3: What is the quotient rule of differentiation?

A3: The quotient rule of differentiation is a rule that states that if $f(x) = \frac{g(x)}{h(x)}$ , then $f^{\prime}(x) = \frac{h(x)g^{\prime}(x) - g(x)h^{\prime}(x)}{[h(x)]^2}$ .

Q4: How do I evaluate the derivative of a function at a specific point?

A4: To evaluate the derivative of a function at a specific point, you can substitute the value of the point into the expression for the derivative.

Q5: What is the significance of derivatives in real-world applications?

A5: Derivatives have numerous applications in various fields, including physics, engineering, economics, and computer science. They are used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of financial markets.

Q6: Can you provide examples of optimization problems?

A6: Yes, here are a few examples of optimization problems:

Maximizing a function: Find the maximum value of a function subject to certain constraints.
Minimizing a function: Find the minimum value of a function subject to certain constraints.
Finding the maximum or minimum of a function subject to constraints: Find the maximum or minimum value of a function subject to certain constraints.

Q7: How do I use derivatives to solve optimization problems?

A7: To use derivatives to solve optimization problems, you can follow these steps:

Define the function: Define the function that you want to optimize.
Find the derivative: Find the derivative of the function.
Set the derivative equal to zero: Set the derivative equal to zero and solve for the input value.
Check the second derivative: Check the second derivative to determine whether the function has a maximum or minimum value.

Q8: Can you provide examples of real-world applications of derivatives?

A8: Yes, here are a few examples of real-world applications of derivatives:

Physics: Derivatives are used to model the motion of objects, such as the trajectory of a projectile or the motion of a pendulum.
Engineering: Derivatives are used to design and optimize systems, such as the design of a bridge or the optimization of a manufacturing process.
Economics: Derivatives are used to model the behavior of financial markets, such as the behavior of stock prices or the behavior of interest rates.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about derivatives and trigonometric functions. We have discussed the derivative of a function, the quotient rule of differentiation, and the significance of derivatives in real-world applications. We have also provided examples of optimization problems and real-world applications of derivatives.