Problem 13 - Calculus for Biology and Medicine

Question

Find \(\frac{d y}{d x}\) if \(y=\arccos x\).

Step-by-Step Solution

Answer

The derivative \(\frac{dy}{dx}\) for \(y = \arccos x\) is \(-\frac{1}{\sqrt{1-x^2}}\).

Step 1: Understand the Derivative of Inverse Functions

We need to find the derivative of the inverse trigonometric function \(y = \arccos x\). First, recall that for \(y = \arccos x\), the derivative is given by the formula \(-\frac{1}{\sqrt{1-x^2}}\).

Step 2: Apply the Formula

Using the derivative formula for \(\arccos x\), substitute it into the expression for \(\frac{d y}{d x}\). This gives us \(-\frac{1}{\sqrt{1-x^2}}\), which is already simplified as the result.

Key Concepts

Understanding Arccosine

The arccosine function, denoted as \( \arccos x \), is the inverse of the cosine function. This function plays a crucial role in trigonometry as it helps find the angle whose cosine value is \( x \). Unlike the simple cosine function that maps angles to values, arccosine takes a value and returns an angle.

• Domain of \( \arccos x \): \([-1, 1] \)
• Range of \( \arccos x \): \([0, \pi] \)

The range restriction ensures that each output corresponds to a unique input, a necessary condition for inverse functions. The arccosine function tends to be more complex when dealing with differentiation due to its limited domain and range.

Derivative Formulas

Derivative formulas are critical tools in calculus that aid in finding the rate at which one variable changes with respect to another. When dealing with inverse trigonometric functions, these formulas become even more important since these functions do not have straightforward definitions like their trigonometric counterparts.

• Derivative of \( \arccos x \): \(-\frac{1}{\sqrt{1-x^2}}\)

This formula arises from the foundational definition of derivatives and the specific properties of the inverse trigonometric functions, like \( \arcsin x \) and \( \arccos x \). When using this derivative, it's also essential to remember that it only holds for \( -1 < x < 1 \), where the expression under the square root doesn’t become negative.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential in bridging the gap between algebra and trigonometry. They allow us to find specific angles when the value of a trigonometric function is known. In addition to \( \arccos x \), common inverse trigonometric functions include \( \arcsin x \), \( \arctan x \), and their counterparts.

• Inverse functions reverse the action of the original trigonometric function.
• They have specific derivatives that aid in calculus, such as in finding the slope of a tangent line.

Understanding these functions is vital because they often appear in numerous calculus problems, especially when differentiating or integrating. They provide a way to transition from a trigonometric evaluation to an analytical expression easily.