Question

Finding a Derivative of a Trigonometric Function. In Exercises \(39-54,\) find the derivative of the trigonometric function. $$ f(x)=-x+\tan x $$

Short answer

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

Step-by-step solution

Identify the Functions and their Rules

The function \(f(x) = -x + \tan x\) is a combination of a linear function and a trigonometric function. Recognize that the derivative of \(x\) is 1, and the derivative of \(\tan x\) is \(\sec^2 x\).

Apply the Sum/Difference Rule

The sum/difference rule states that the derivative of a sum of functions is the sum of the derivatives. Thus, calculate the derivative of each part separately: the derivative of \(-x\) and the derivative of \(\tan x\).

Compute the Derivatives

The derivative of \(-x\) is \(-1\), and the derivative of \(\tan x\) is \(\sec^2 x\). Therefore, by the sum rule, the derivative of the function \(f(x) = -x + \tan x\) is \(-1 + \sec^2 x\).

Key concepts

Sum/Difference Rule

The sum/difference rule in calculus is a fundamental principle that helps simplify the differentiation process of combined functions. It states that the derivative of a sum or difference of functions is simply the sum or difference of their individual derivatives. This rule is essential when dealing with functions like the example function, \(f(x) = -x + \tan x\), where you have to find the derivatives of each part separately.
For instance, consider the function \(g(x) = h(x) + k(x)\). According to the sum/difference rule, the derivative \(g'(x)\) is given by \(h'(x) + k'(x)\).
Similarly, if you have a function \(m(x) = p(x) - q(x)\), the derivative \(m'(x)\) will be \(p'(x) - q'(x)\). By using this rule, the often complex task of differentiation becomes more manageable.

Derivative of Linear Functions

The derivative of linear functions can be understood as the rate of change or the slope of the line. A linear function is generally of the form \(ax + b\), where \(a\) is the slope and \(b\) is the y-intercept.
The derivative of a simple linear function, \(f(x) = x\), is constant and is given as 1 since the rate of change remains the same along the line.
If there's a negative sign involved, such as in \(-x\), the derivative becomes \(-1\). This signifies the steepness and direction of the line - in this case, a steeper downward slope.
Thus, knowing how to differentiate linear functions efficiently using this basic understanding can significantly simplify more compound differentiation problems.

Trigonometric Differentiation

Trigonometric differentiation involves finding the derivatives of trigonometric functions like sine, cosine, and tangent. Each of these has specific derivative formulas that are crucial in calculus.
For example, consider the function \(\tan x\). Its derivative is \(\sec^2 x\). This emerges from its properties and can be derived using the quotient rule due to its definition as \(\frac{\sin x}{\cos x}\).
Trigonometric differentiation requires you to memorize certain derivatives:

• The derivative of \(\sin x\) is \(\cos x\)
• The derivative of \(\cos x\) is \(-\sin x\)
• The derivative of \(\tan x\) is \(\sec^2 x\)

Mastery of these core derivatives facilitates the process of solving complex problems involving trigonometric functions.

Calculus Fundamentals

Calculus is the mathematical study of change and includes essential principles such as differentiation and integration. Differentiation focuses on small changes and rates of change, which help determine functions' behavior.
A central component of calculus is understanding how one quantity changes with respect to another, like how the change in time affects speed in physics. It starts from basic principles such as limits, which involve finding the value that a function approaches as the input approaches some point.
Differentiation is the process of finding the derivative, which quantifies how a function changes at any given point with respect to another variable. This fundamental concept enables us to understand and predict the behavior of dynamic systems.
By comprehending these concepts, calculus becomes not only a tool for theoretical mathematics but a practical solution for real-world problems.