Derivative of Trigonometric Functions

Oct 13, 2025 by windailys.com 31 views

If $f(x) = 2 \sin x + 8 \cos x$ , then: Calculating Derivatives and Function Values

Introduction

In calculus, the 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 derivative of the function $f(x) = 2 \sin x + 8 \cos x$ and calculate its value at a specific point.

Calculating the Derivative of $f(x)$

To find the derivative of $f(x)$ , we will use the sum rule and the chain rule of differentiation. The sum rule states that the derivative of a sum is the sum of the derivatives, while the chain rule states that the derivative of a composite function is the product of the derivatives of the outer and inner functions.

Using the chain rule, we can write the derivative of $f(x)$ as:

f^{\prime}(x) = \frac{d}{dx} (2 \sin x + 8 \cos x)

f^{\prime}(x) = 2 \frac{d}{dx} (\sin x) + 8 \frac{d}{dx} (\cos x)

f^{\prime}(x) = 2 \cos x - 8 \sin x

Calculating $f^{\prime}(3)$

Now that we have found the derivative of $f(x)$ , we can calculate its value at $x = 3$ .

f^{\prime}(3) = 2 \cos 3 - 8 \sin 3

To evaluate this expression, we need to find the values of $\cos 3$ and $\sin 3$ . Using a calculator or a trigonometric table, we can find that:

\cos 3 \approx -0.990

\sin 3 \approx -0.142

Substituting these values into the expression for $f^{\prime}(3)$ , we get:

f^{\prime}(3) \approx 2 (-0.990) - 8 (-0.142)

f^{\prime}(3) \approx -1.980 + 1.136

f^{\prime}(3) \approx -0.844

Conclusion

In this article, we have calculated the derivative of the function $f(x) = 2 \sin x + 8 \cos x$ and found its value at $x = 3$ . The derivative of $f(x)$ is given by $f^{\prime}(x) = 2 \cos x - 8 \sin x$ , and its value at $x = 3$ is approximately $-0.844$ . This result demonstrates the importance of derivatives in calculus and their applications in various fields.

Derivative of Trigonometric Functions

The derivative of a trigonometric function is a fundamental concept in calculus. In this section, we will explore the derivatives of the six basic trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant.

Derivative of Sine

The derivative of the sine function is the cosine function:

\frac{d}{dx} (\sin x) = \cos x

Derivative of Cosine

The derivative of the cosine function is the negative sine function:

\frac{d}{dx} (\cos x) = -\sin x

Derivative of Tangent

The derivative of the tangent function is the secant squared function:

\frac{d}{dx} (\tan x) = \sec^2 x

Derivative of Cotangent

The derivative of the cotangent function is the negative cosecant squared function:

\frac{d}{dx} (\cot x) = -\csc^2 x

Derivative of Secant

The derivative of the secant function is the secant tangent function:

\frac{d}{dx} (\sec x) = \sec x \tan x

Derivative of Cosecant

The derivative of the cosecant function is the negative cosecant cotangent function:

\frac{d}{dx} (\csc x) = -\csc x \cot x

Applications of Derivatives

Derivatives have numerous applications in various fields, including physics, engineering, and economics. In this section, we will explore some of the most important applications of derivatives.

Physics

Derivatives are used to describe the motion of objects in physics. For example, the derivative of the position function represents the velocity of an object, while the derivative of the velocity function represents the acceleration of an object.

Engineering

Derivatives are used to design and optimize systems in engineering. For example, the derivative of the cost function represents the rate of change of the cost with respect to the input, which can be used to optimize the design of a system.

Economics

Derivatives are used to model and analyze economic systems in economics. For example, the derivative of the demand function represents the rate of change of the demand with respect to the price, which can be used to analyze the behavior of consumers.

Conclusion

In conclusion, derivatives are a fundamental concept in calculus and have numerous applications in various fields. The derivative of a function represents the rate of change of the function with respect to its input, and it is used to describe the motion of objects in physics, design and optimize systems in engineering, and model and analyze economic systems in economics.
Derivatives: A Comprehensive Guide to Calculus

Q&A: Derivatives

Q1: What is a derivative?

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

Q2: How do you calculate the derivative of a function?

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

The derivative of a constant is 0.
The derivative of a sum is the sum of the derivatives.
The derivative of a product is the derivative of the first function times the second function plus the first function times the derivative of the second function.
The derivative of a quotient is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Q3: What is the derivative of the sine function?

A3: The derivative of the sine function is the cosine function.

Q4: What is the derivative of the cosine function?

A4: The derivative of the cosine function is the negative sine function.

Q5: How do you use derivatives in real-life applications?

A5: Derivatives are used in a variety of real-life applications, including:

Physics: to describe the motion of objects
Engineering: to design and optimize systems
Economics: to model and analyze economic systems
Computer Science: to optimize algorithms and solve problems

Q6: What is the difference between a derivative and a differential?

A6: A derivative is a measure of how a function changes as its input changes, while a differential is a measure of the change in the function itself.

Q7: How do you calculate the derivative of a function with multiple variables?

A7: To calculate the derivative of a function with multiple variables, you can use the following rules:

The derivative of a function with multiple variables is the sum of the partial derivatives of each variable.
The partial derivative of a function with respect to a variable is the derivative of the function with respect to that variable, while holding all other variables constant.

Q8: What is the chain rule in calculus?

A8: The chain rule is a rule in calculus that allows you to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Q9: How do you use the chain rule in real-life applications?

A9: The chain rule is used in a variety of real-life applications, including:

Physics: to describe the motion of objects
Engineering: to design and optimize systems
Economics: to model and analyze economic systems
Computer Science: to optimize algorithms and solve problems

Q10: What is the fundamental theorem of calculus?

A10: The fundamental theorem of calculus is a theorem that relates the derivative of a function to the definite integral of the function. It states that the definite integral of a function is equal to the antiderivative of the function evaluated at the upper limit minus the antiderivative of the function evaluated at the lower limit.

Conclusion

In conclusion, derivatives are a fundamental concept in calculus and have numerous applications in various fields. Understanding derivatives is essential for solving problems in physics, engineering, economics, and computer science. By mastering the rules and techniques of derivatives, you can unlock the secrets of calculus and apply it to real-life problems.

Additional Resources

Calculus textbooks: "Calculus" by Michael Spivak, "Calculus" by James Stewart
Online resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha
Calculus software: Mathematica, Maple, MATLAB

Final Thoughts

Derivatives are a powerful tool in calculus that can be used to solve a wide range of problems. By mastering the rules and techniques of derivatives, you can unlock the secrets of calculus and apply it to real-life problems. Remember to practice regularly and seek help when needed to become proficient in derivatives.