Question:
Derivative of \( \cos^{-1}\left( \frac{\sin x + \cos x}{\sqrt{2}} \right) \), \( -\frac{\pi}{4} < x < \frac{\pi}{4} \) with respect to \( x \) is :
Derivative of \( \cos^{-1}\left( \frac{\sin x + \cos x}{\sqrt{2}} \right) \), \( -\frac{\pi}{4} < x < \frac{\pi}{4} \) with respect to \( x \) is :
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Always double-check interval ranges. Since \( \cos(- \theta) = \cos(\theta) \), adapting the sign of the inside angle ensures compliance with the principal range of \( \cos^{-1} \).
Updated On: Jul 6, 2026
- \( -1 \)
- \( 1 \)
- \( \frac{\pi}{4} \)
- \( -\frac{\pi}{4} \)
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The Correct Option is A
Solution and Explanation
Concept:
To differentiate complex expressions involving inverse trigonometric functions, simplify the inner argument using standard trigonometric identities before computing the derivative.
Step 1: Simplify the inner expression. Let \( y = \cos^{-1}\left( \frac{\sin x + \cos x}{\sqrt{2}} \right) \). We can distribute the denominator: \[ \frac{\sin x + \cos x}{\sqrt{2}} = \frac{1}{\sqrt{2}}\cos x + \frac{1}{\sqrt{2}}\sin x \] Recall the standard trigonometric values: \( \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \) and \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \). Substituting these into the formula: \[ \frac{\sin x + \cos x}{\sqrt{2}} = \cos x \cos\left(\frac{\pi}{4}\right) + \sin x \sin\left(\frac{\pi}{4}\right) \] Using the cosine subtraction identity, \( \cos(A - B) = \cos A \cos B + \sin A \sin B \), we rewrite it as: \[ \cos\left(x - \frac{\pi}{4}\right) \]
Step 2: Substitute back into the inverse function. \[ y = \cos^{-1}\left( \cos\left(x - \frac{\pi}{4}\right) \right) \] We are given the domain restriction: \( -\frac{\pi}{4} < x < \frac{\pi}{4} \). Subtracting \( \frac{\pi}{4} \) across the inequality: \[ -\frac{\pi}{4} - \frac{\pi}{4} < x - \frac{\pi}{4} < \frac{\pi}{4} - \frac{\pi}{4} \implies -\frac{\pi}{2} < x - \frac{\pi}{4} < 0 \] Since cosine is an even function, \( \cos\left(x - \frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4} - x\right) \). Under our domain, the angle \( \left(\frac{\pi}{4} - x\right) \) lies in \( \left(0, \frac{\pi}{2}\right) \), which is well within the principal value branch of \( \cos^{-1} \) (i.e., \( [0, \pi] \)). Thus: \[ y = \cos^{-1}\left( \cos\left(\frac{\pi}{4} - x\right) \right) = \frac{\pi}{4} - x \]
Step 3: Differentiate with respect to \( x \). Now, we take the derivative of our simplified equation: \[ \frac{dy}{dx} = \frac{d}{dx}\left( \frac{\pi}{4} - x \right) = 0 - 1 = -1 \]
Step 1: Simplify the inner expression. Let \( y = \cos^{-1}\left( \frac{\sin x + \cos x}{\sqrt{2}} \right) \). We can distribute the denominator: \[ \frac{\sin x + \cos x}{\sqrt{2}} = \frac{1}{\sqrt{2}}\cos x + \frac{1}{\sqrt{2}}\sin x \] Recall the standard trigonometric values: \( \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \) and \( \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \). Substituting these into the formula: \[ \frac{\sin x + \cos x}{\sqrt{2}} = \cos x \cos\left(\frac{\pi}{4}\right) + \sin x \sin\left(\frac{\pi}{4}\right) \] Using the cosine subtraction identity, \( \cos(A - B) = \cos A \cos B + \sin A \sin B \), we rewrite it as: \[ \cos\left(x - \frac{\pi}{4}\right) \]
Step 2: Substitute back into the inverse function. \[ y = \cos^{-1}\left( \cos\left(x - \frac{\pi}{4}\right) \right) \] We are given the domain restriction: \( -\frac{\pi}{4} < x < \frac{\pi}{4} \). Subtracting \( \frac{\pi}{4} \) across the inequality: \[ -\frac{\pi}{4} - \frac{\pi}{4} < x - \frac{\pi}{4} < \frac{\pi}{4} - \frac{\pi}{4} \implies -\frac{\pi}{2} < x - \frac{\pi}{4} < 0 \] Since cosine is an even function, \( \cos\left(x - \frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4} - x\right) \). Under our domain, the angle \( \left(\frac{\pi}{4} - x\right) \) lies in \( \left(0, \frac{\pi}{2}\right) \), which is well within the principal value branch of \( \cos^{-1} \) (i.e., \( [0, \pi] \)). Thus: \[ y = \cos^{-1}\left( \cos\left(\frac{\pi}{4} - x\right) \right) = \frac{\pi}{4} - x \]
Step 3: Differentiate with respect to \( x \). Now, we take the derivative of our simplified equation: \[ \frac{dy}{dx} = \frac{d}{dx}\left( \frac{\pi}{4} - x \right) = 0 - 1 = -1 \]
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