Dickson Bowman
11/08/2023 · Elementary School
4 points \( f(x)=\cos \left(\cos ^{-1} x\right) \), find \( f^{\prime}(x) \). You must simplify the result. If you select an intermediate solution not the simplified solution, it is wrong.
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Tutor-Verified Answer
Step-by-step Solution
Find the first order derivative with respect to \( x \) for \( \cos(\arccos(x)) \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dx}\left(\cos\left(\arccos\left(x\right)\right)\right)\)
- step1: Use differentiation rules:
\(\frac{d}{dg}\left(\cos\left(g\right)\right)\times \frac{d}{dx}\left(\arccos\left(x\right)\right)\)
- step2: Find the derivative:
\(-\sin\left(g\right)\times \frac{d}{dx}\left(\arccos\left(x\right)\right)\)
- step3: Calculate:
\(-\sin\left(g\right)\left(-\frac{1}{\sqrt{1-x^{2}}}\right)\)
- step4: Substitute back:
\(-\sin\left(\arccos\left(x\right)\right)\left(-\frac{1}{\sqrt{1-x^{2}}}\right)\)
- step5: Calculate:
\(-\sqrt{1-x^{2}}\times \left(-\frac{1}{\sqrt{1-x^{2}}}\right)\)
- step6: Multiply the terms:
\(\sqrt{1-x^{2}}\times \frac{1}{\sqrt{1-x^{2}}}\)
- step7: Reduce the fraction:
\(1\)
The derivative of the function \( f(x)=\cos(\arccos(x)) \) with respect to \( x \) is \( f'(x) = 1 \).
Quick Answer
\( f'(x) = 1 \)
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