Problem 115 - Thomas Calculus in SI Units

Question

Use the identity \(\csc ^{-1} u=\frac{\pi}{2}-\sec ^{-1} u\) to derive the formula for the derivative of \(\csc ^{-1} u\) in Table 7.3 from the formula for the derivative of \(\sec ^{-1} u\).

Step-by-Step Solution

Answer

The derivative of \( \csc^{-1} u \) is \( -\frac{1}{|u|\sqrt{u^2 - 1}} \).

Step 1: Understand the Given Identity

The identity provided: \( \csc ^{-1} u = \frac{\pi}{2} - \sec ^{-1} u \) allows us to express the inverse cosecant function in terms of the inverse secant function. This relationship can be leveraged to find the derivative of \( \csc^{-1} u \) if we already know the derivative of \( \sec^{-1} u \).

Step 2: Differentiate Both Sides with Respect to u

To find the derivative \( \frac{d}{du}(\csc^{-1} u) \), differentiate both sides of the identity \( \csc^{-1} u = \frac{\pi}{2} - \sec^{-1} u \): \[ \frac{d}{du}(\csc^{-1} u) = 0 - \frac{d}{du}(\sec^{-1} u) \] This simplifies to: \[ \frac{d}{du}(\csc^{-1} u) = - \frac{d}{du}(\sec^{-1} u) \].

Step 3: Use the Known Derivative of \( \sec^{-1} u \)

Recall the formula for the derivative of \( \sec^{-1} u \): \[ \frac{d}{du}(\sec^{-1} u) = \frac{1}{|u|\sqrt{u^2 - 1}} \] Using this, substitute into the equation from Step 2: \[ \frac{d}{du}(\csc^{-1} u) = - \frac{1}{|u|\sqrt{u^2 - 1}} \] This gives us the derivative of \( \csc^{-1} u \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential in calculus and are used to reverse or 'undo' trigonometric functions like sine, cosine, and tangent. In contexts where you know the value of a trigonometric function and need to find the corresponding angle, inverse trigonometric functions come in handy.
Let's take a brief look at some common ones:

• \( ext{Arcsin} \) or \( ext{sin}^{-1} \) returns the angle whose sine is a given number.
• \( ext{Arccos} \) or \( ext{cos}^{-1} \) gives the angle whose cosine is a specified number.
• \( ext{Arctan} \) or \( ext{tan}^{-1} \) provides the angle whose tangent is the input.
• The less commonly used functions in basic trigonometry include \( ext{Arccsc} \), \( ext{Arcsec} \), and \( ext{Arccot} \), the inverses of the cosecant, secant, and cotangent functions respectively.

In the given problem, we deal with the inverse secant \( \sec^{-1} \) and inverse cosecant \( \csc^{-1} \), tying them together with an identity to derive their derivatives. This step links our understanding of inverse trigonometric functions with other calculus concepts.

Derivative of Inverse Functions

The derivative of inverse functions is critical for determining how the inverse function's output changes with respect to its input. For the inverse trigonometric functions specifically, finding derivatives can initially seem daunting, but utilizing known identities simplifies the process.
In our example, the identity \( \csc ^{-1} u = \frac{\pi}{2} - \sec ^{-1} u \) allows expressing \( \csc^{-1} u \) in terms of \( \sec^{-1} u \). When differentiating using this relationship, both sides are derived with respect to \( u \):

• The derivative of \( \sec^{-1} u \) is known to be \( \frac{1}{|u|\sqrt{u^2 - 1}} \).
• Thus, by differentiating the identity, the derivative of \( \csc^{-1} u \) is obtained as \( - \frac{1}{|u|\sqrt{u^2 - 1}} \).

This technique showcases how calculus can be used creatively to link related functions through their identities, streamlining what might otherwise be an arduous calculation.

Identity in Calculus

In calculus, identities serve as powerful tools to bridge function types and transform complex functions into more manageable forms. They play a key role in simplifying the calculation of derivatives and integrals and are pivotal in the manipulation of mathematical expressions.
Identities can be equations like the one used in the exercise \( \csc^{-1} u = \frac{\pi}{2} - \sec^{-1} u \). This identity is a simplified form that links \( \sec \) and \( \csc \) functions. It intuitively unravels each function's behavior by relating them through standard trigonometric properties.
Moreover, identities don't just stay with trigonometric functions. They're omnipresent in calculus, including polynomial identities, logarithmic identities, and complex number identities. Often, an identity connects different mathematical ideas by highlighting a deeper, invariant feature common to all expressions involved.In summary, understanding and employing identities in calculus frequently leads to a more efficient and elegant solution, turning potentially intricate derivative problems into a straightforward substitution of known results.