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Chapter 2: Problem 16

Find \(f^{\prime}(x)\). $$ f(x)=\sec ^{2} x-\tan ^{2} x $$

Short Answer

Expert verified
The derivative of the function is 0.

Step by step solution

01

Apply the Derivative Rules

To find the derivative of the function, we need to apply the derivative rules to each term separately. The function is given by:\[ f(x) = \sec^2 x - \tan^2 x \]We will use the following derivative rules:- The derivative of \(\sec^2 x\) is \(2\sec^2 x \tan x\).- The derivative of \(\tan^2 x\) is \(2\tan x \sec^2 x\) using the chain rule.
02

Differentiate Each Term

Differentiate each term of the function using the rules from Step 1:- For \(\sec^2 x\), the derivative is: \[ \frac{d}{dx}(\sec^2 x) = 2\sec^2 x \tan x \]- For \(\tan^2 x\), the derivative is: \[ \frac{d}{dx}(\tan^2 x) = 2\tan x \sec^2 x \]
03

Combine the Derivatives

Now, combine the derivatives of the individual terms to get the derivative of the entire function:\[ f^{\prime}(x) = 2\sec^2 x \tan x - 2\tan x \sec^2 x \]
04

Simplify the Expression

Notice that both terms in \(f^{\prime}(x)\) are similar and cancel each other out:\[ f^{\prime}(x) = 2\sec^2 x \tan x - 2\sec^2 x \tan x = 0 \]Therefore, the derivative of the function simplifies to zero.

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Key Concepts

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

Trigonometric Functions
Trigonometric functions are foundational mathematical functions that relate the angles of a triangle to the lengths of its sides. In calculus, they play a crucial role when differentiating and integrating functions that involve angles or periodic phenomena. Some of the most common trigonometric functions are:
  • Sine (\(\sin x\)): the ratio of the opposite side to the hypotenuse in a right triangle.
  • Cosine (\(\cos x\)): the ratio of the adjacent side to the hypotenuse.
  • Tangent (\(\tan x\)): the ratio of the opposite side to the adjacent side, or equivalently, \(\frac{\sin x}{\cos x}\).
  • Secant (\(\sec x\)): the reciprocal of cosine, or \(\frac{1}{\cos x}\).
  • Cosecant (\(\csc x\)): the reciprocal of sine, or \(\frac{1}{\sin x}\).
  • Cotangent (\(\cot x\)): the reciprocal of tangent, or \(\frac{1}{\tan x}\).
These functions are cyclical and repeat their values in regular intervals, making them extremely useful in modeling repeating phenomena such as sound waves or the motion of celestial bodies. The derivatives of trigonometric functions are also important, especially in calculus, as they help to analyze the rate of change of these periodic phenomena.
Chain Rule
The chain rule is a fundamental differentiation technique used to find the derivative of a composite function. A composite function is a function made by combining two or more functions, such as \(f(g(x))\). The chain rule states that to differentiate such a function, we need to:
  • Differentiate the outer function (\(f\)) with respect to the inner function (\(g\)).
  • Multiply the result by the derivative of the inner function (\(g(x)\)).
In mathematical terms, if \(h(x) = f(g(x))\), then the derivative \(h'(x)\) is given by \( h'(x) = f'(g(x)) \cdot g'(x) \).
In the exercise presented, the chain rule is applied to \(\tan^2 x\). The tangent function itself needs to be differentiated first, followed by applying the power rule to find the derivative of \(\tan^2 x\). By recognizing the chain of functions here, we are able to calculate the derivative correctly.
Differentiation Techniques
Differentiation techniques are strategies used in calculus to find the derivative of functions, which tells us how a function changes at any given point. These techniques can range from basic rules to more complex applications like the chain rule or product rule.
Some of the basic differentiation rules include:
  • Power Rule: If \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).
  • Constant Rule: The derivative of a constant is zero.
  • Sum Rule: The derivative of a sum of functions is the sum of their derivatives.
  • Difference Rule: The derivative of a difference of functions is the difference of their derivatives.
In the given exercise, specific rules for differentiating trigonometric functions were used:
  • The derivative of \(\sec^2 x\) is \(2\sec^2 x \tan x\), known from the chain rule and specific identities.
  • The derivative of \(\tan^2 x\) involves using the chain rule, first identifying \(\tan x\) as an inner function.
Using the correct differentiation technique allows for accurate calculation of derivatives, demonstrating changes, rates, and trends effectively in functions expressing real-world phenomena.

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Most popular questions from this chapter

Determine whether the statement is true or false. Explain your answer. If \(y=f(x)\), then \(\frac{d}{d x}[\sqrt{y}]=\sqrt{f^{\prime}(x)}\).

Find \(f^{\prime}(x)\). \(f(x)=\left[x+\csc \left(x^{3}+3\right)\right]^{-3}\)

Find all points where \(f\) fails to be differentiable. Justify your answer. (a) \(f(x)=|3 x-2|\) (b) \(f(x)=\left|x^{2}-4\right|\)

You are asked in these exercises to determine whether a piecewise-defined function \(f\) is differentiable at a value \(x=x_{0}\) where \(f\) is defined by different formulas on different sides of \(x_{0} .\) You may use without proof the following result, which is a consequence of the Mean-Value Theorem (discussed in Section 4.8). Theorem. Let \(f\) be continuous at \(x_{0}\) and suppose that \(\lim _{x \rightarrow x_{0}} f^{\prime}(x)\) exists. Then \(f\) is differentiable at \(x_{0}\), and \(f^{\prime}\left(x_{0}\right)=\lim _{x \rightarrow x_{0}} f^{\prime}(x)\) Let $$ f(x)=\left\\{\begin{array}{ll} x^{2}, & x \leq 1 \\ \sqrt{x}, & x>1 \end{array}\right. $$ Determine whether \(f\) is differentiable at \(x=1\). If so, find the value of the derivative there.

The force \(F\) (in pounds) acting at an angle \(\theta\) with the horizontal that is needed to drag a crate weighing \(W\) pounds along a horizontal surface at a constant velocity is given by $$ F=\frac{\mu W}{\cos \theta+\mu \sin \theta} $$ where \(\mu\) is a constant called the coefficient of sliding friction between the crate and the surface (see the accompanying figure). Suppose that the crate weighs \(150 \mathrm{lb}\) and that \(\mu=0.3\) (a) Find \(d F / d \theta\) when \(\theta=30^{\circ} .\) Express the answer in units of pounds/degree. (b) Find \(d F / d t\) when \(\theta=30^{\circ}\) if \(\theta\) is decreasing at the rate of \(0.5^{\circ} / \mathrm{s}\) at this instant.
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