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Chapter 2: Problem 43

Find the derivative of the trigonometric function. \(f(x)=-x+\tan x\)

Short Answer

Expert verified
The derivative of the given function \(f(x)=-x+\tan x\) is \(-1 + \sec^2 x\).

Step by step solution

01

Differentiate the linear function

To differentiate the linear function -x, you should know that the derivative of x with respect to x is 1. Therefore, considering the sign, the derivative of -x will be -1.
02

Differentiate the trigonometric function

To differentiate the trigonometric function \(\tan x\), you should recognize that the derivative of \(\tan x\) with respect to x is \(\sec^2 x\).
03

Combine the results

Combine the two derivatives to yield the final derivative of the given function. Simply take the derivative of each term and keep their signs as they were.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differentiation Rules
In calculus, differentiation is all about finding the rate at which a function is changing at any given point and is fundamental to calculus. Knowing the differentiation rules helps in finding derivatives easily. For any function that is a sum or difference of terms, like in the original problem, you can differentiate each term individually and then add or subtract these derivatives.

Key rules include:
  • Constant Rule: The derivative of a constant is zero.
  • Power Rule: The derivative of \( x^n \) is \( nx^{n-1} \).
  • Sum/Difference Rule: The derivative of a sum/difference is simply the sum/difference of their derivatives.
For the function \(-x + \tan x\), we apply the Sum/Difference Rule by differentiating \(-x\) and \(\tan x\) separately and then combining them. Understanding these rules helps simplify complicated problems by breaking them into manageable parts.
Trigonometric Derivatives
A deep grasp of trigonometric derivatives is essential for working with functions that include trigonometric terms. Trigonometric derivatives are formulas that specify how to differentiate trigonometric functions.

For the basic trigonometric functions, you should know:
  • \( \frac{d}{dx} (\sin x) = \cos x \)
  • \( \frac{d}{dx} (\cos x) = -\sin x \)
  • \( \frac{d}{dx} (\tan x) = \sec^2 x \)
In the exercise provided, we encounter \(\tan x\). By knowing that the derivative of \(\tan x\) is \(\sec^2 x\), you can quickly calculate this part of the function's overall derivative. Remember, each trigonometric function has a specific pattern that helps in immediate recognition and differentiation.
Calculus Concepts
At the heart of calculus is the study of change, which often involves finding how one quantity changes in relation to another. Differentiation is one major area of calculus. It involves computing the derivative, which in simple terms is the slope of the tangent to the function at any point.

To understand why differentiation is so useful, consider its real-world applications:
  • Rate of Change: Derivatives can show how a quantity changes over time, like velocity and acceleration in physics.
  • Optimization: By finding where the derivative is zero, you can determine local maxima and minima, helping to optimize processes.
  • Linear Approximations: Derivatives help approximate values of complicated functions.
Applying calculus concepts to differentiate \(-x + \tan x\), you get \(-1 + \sec^2 x\). This result tells us how the given function changes in small increments of \(x\). As you dive deeper, these foundational calculus concepts enable you to tackle more complex functions easily.

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Most popular questions from this chapter

Prove (Theorem 2.3) that \(\frac{d}{d x}\left[x^{n}\right]=n x^{n-1}\) for the case in which \(n\) is a rational number. (Hint: Write \(y=x^{p / q}\) in the form \(y^{q}=x^{p}\) and differentiate implicitly. Assume that \(p\) and \(q\) are integers, where \(q>0 .\) )

Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. $$ f(x)=\frac{1}{\sqrt{x+4}}, \quad\left(0, \frac{1}{2}\right) $$

Verify that the two families of curves are orthogonal where \(C\) and \(K\) are real numbers. Use a graphing utility to graph the two families for two values of \(C\) and two values of \(K\). $$ x^{2}+y^{2}=C^{2}, \quad y=K x $$

Determine the point(s) in the interval \((0,2 \pi)\) at which the graph of \(f(x)=2 \cos x+\sin 2 x\) has a horizontal tangent.

Find the second derivative of the function. $$ f(x)=\sec ^{2} \pi x $$
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