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Magazine Chapter 4: Problem 1
Show that the derivative of \(\arccos x\) is \(-\frac{1}{\sqrt{1-x^{2}}}\).
Short Answer
Expert verified
The derivative of \( \arccos x \) is \(-\frac{1}{\sqrt{1-x^{2}}}\).
Step by step solution
01
Set up the function
We want to find the derivative of the function \( y = \arccos x \). This inverse trigonometric function implies \( \cos y = x \).
02
Implicit differentiation
Differentiate both sides of \( \cos y = x \) with respect to \( x \). Using the chain rule, the derivative of \( \cos y \) with respect to \( y \) gives \(-\sin y \), and then multiply by \( \frac{dy}{dx} \): \(-\sin y \cdot \frac{dy}{dx} = 1\).
03
Solve for \( \frac{dy}{dx} \)
Rearrange the equation from the previous step to solve for \( \frac{dy}{dx} \): \(\frac{dy}{dx} = -\frac{1}{\sin y}\).
04
Express \( \sin y \) in terms of \( x \)
Since \( \cos y = x \), by the Pythagorean identity, \( \sin^2 y = 1 - \cos^2 y \). Thus, \( \sin y = \sqrt{1 - x^2} \).
05
Substitute \( \sin y \) back into the derivative
Substitute \( \sin y = \sqrt{1 - x^2} \) into the expression for \( \frac{dy}{dx} \): \(\frac{dy}{dx} = -\frac{1}{\sqrt{1 - x^2}}\).
06
Conclude the result
Thus, the derivative of \( \arccos x \) with respect to \( x \) is \(-\frac{1}{\sqrt{1 - x^2}}\). This confirms the originally given derivative formula.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Implicit Differentiation
When we differentiate both sides of the equation \( \cos y = x \), we are applying **implicit differentiation**. This method is essential when dealing with equations where one variable is a function of another, like in inverse trigonometric functions. Here, \( y = \arccos x \) is an implicit relationship, making direct differentiation difficult.
To differentiate \( \cos y \) with respect to \( x \), treat \( y \) as a function of \( x \), similar to the word 'implicit' meaning 'understood' or 'assumed'.
To differentiate \( \cos y \) with respect to \( x \), treat \( y \) as a function of \( x \), similar to the word 'implicit' meaning 'understood' or 'assumed'.
- Differentiate both sides of the equation, keeping in mind that \( y \) is a function of \( x \).
- This involves taking the derivative of \( \cos y \) using the chain rule and introducing \( \frac{dy}{dx} \) since \( y \) indirectly depends on \( x \).
Chain Rule
The **chain rule** is a fundamental tool in calculus for differentiating composite functions. When you encounter a function within another function, like \( \cos y \), which contains \( y \) that depends on \( x \), you use the chain rule to differentiate correctly.
To apply the chain rule:
To apply the chain rule:
- Differentiate the outer function \( \cos y \) as if \( y \) were a separate variable, leading to \(-\sin y \).
- Multiply the result by the derivative of the inner function, \( \frac{dy}{dx} \), to account for \( y \)'s dependence on \( x \).
Pythagorean Identity
The **Pythagorean identity** is a key trigonometric principle that relates the sine and cosine of an angle. For any angle \( y \) in a right triangle, this identity states: \[\sin^2 y + \cos^2 y = 1\]
In solving the given derivative problem, we start with \( \cos y = x \) and need \( \sin y \) to substitute back into our derivative expression. Here鈥檚 how the identity helps:
In solving the given derivative problem, we start with \( \cos y = x \) and need \( \sin y \) to substitute back into our derivative expression. Here鈥檚 how the identity helps:
- Rearrange the Pythagorean identity to solve for \( \sin^2 y \): \( \sin^2 y = 1 - \cos^2 y \).
- Substitute \( \cos y = x \) to find \( \sin^2 y = 1 - x^2 \).
- Take the square root to express \( \sin y \) in its simplest form: \( \sin y = \sqrt{1 - x^2} \).
