Question

Finding a Derivative of a Trigonometric Function. In Exercises \(39-54,\) find the derivative of the trigonometric function. $$ h(x)=\frac{1}{x}-12 \sec x $$

Short answer

The derivative of \(h(x) = 1/x - 12 sec x \) is \(h'(x) = -1/x^2 - 12 sec(x) tan(x) \).

Step-by-step solution

Identify and Separate the Functions

The function \(h(x) = 1/x - 12 sec x \) is composed of two functions subtracted: \(f(x) = 1/x\) and \(g(x) = -12 sec x\). The derivative will be found by taking the derivative of each function separately and then combining the results.

Apply the Quotient Rule for the first function

First, let's take the derivative of \(f(x) = 1/x\). We can rewrite it as \(x^{-1}\) to apply the power rule. Using the power rule, the derivative of \( x^n \) is \( n*x^{n-1} \). That gives us \(-x^{-2}\) or \(-1/x^2\).

Find the Derivative of the Trigonometric Function

Now, let's find the derivative of \(g(x) = -12 sec x\). The derivative of \( sec(x) \) is \( sec(x)*tan(x) \). So, when we multiply this by -12, we get \(-12* sec(x)*tan(x)\) as the derivative.

Combine the Derivatives

Finally, subtract \(g'(x) = -12 sec(x) tan(x)\) from \(f'(x) = -1/x^2\) to get the final derivative of \(h'(x)\). Thus, the derivative of the original function, \(h'(x) = -1/x^2 - 12 sec(x) tan(x) \).

Key concepts

Trigonometric Functions

Trigonometric functions are mathematical functions related to angles and right-angled triangles. The most common trigonometric functions are sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). They relate the angles of a triangle to the lengths of its sides. These functions are fundamental in the study of periodic phenomena, such as waves and oscillations.

In the context of differentiation, another important trigonometric function is the secant function, written as \( \sec \theta \). This is simply the reciprocal of the cosine function: \( \sec \theta = \frac{1}{\cos \theta} \). Trigonometric functions are often encountered in calculus when dealing with periodic or wave-like behavior.

Power Rule

The power rule is a basic rule for differentiating functions of the form \( x^n \), where \( n \) is any real number. It provides a quick and effective way to find the derivative. The rule states that if you have \( f(x) = x^n \), then the derivative, \( f'(x) \), is \( n*x^{n-1} \).

In our exercise, this rule helps simplify the differentiation of \( 1/x \). By rewriting \( 1/x \) as \( x^{-1} \), the power rule can be applied to find the derivative easily. Thus, \( \frac{d}{dx}(x^{-1}) = -1*x^{-2} = -\frac{1}{x^2} \). The simplicity of the power rule makes it a staple in calculus for handling polynomial terms.

Quotient Rule

The quotient rule is used for finding the derivative of a quotient of two functions, where one function is divided by another. This rule states that for functions \( u(x) \) and \( v(x) \), the derivative \( \frac{d}{dx}\left(\frac{u}{v}\right) \) is given by:\[ \frac{u'v - uv'}{v^2} \]where \( u' \) and \( v' \) are the derivatives of \( u \) and \( v \), respectively.

In our case, the derivative of \( f(x) = \frac{1}{x} \), which can also be approached by rewriting it as \( x^{-1} \) to conveniently apply the power rule. However, understanding the quotient rule remains valuable for handling more complex functions where simple rewriting might not be possible.

Secant Function

The secant function, denoted as \( \sec(x) \), is an important trigonometric function. It is the reciprocal of the cosine function, expressed as \( \sec(x) = \frac{1}{\cos(x)} \). In calculus, it's important to know the derivative of secant, which is \( \sec(x) \tan(x) \).

When working with derivatives involving secant, like in our exercise, recognizing and applying this derivative is crucial. For \( g(x) = -12 \sec(x) \), the derivative becomes \( -12 \sec(x) \tan(x) \). Understanding these relationships allows for easier manipulation of expressions involving secant functions in calculus.