Question:
If \(f(x)=\dfrac{\sin x+\cos x}{\sin x-\cos x}\), the value of \(f'(x)\) at \(x=0\) is ________________.
If \(f(x)=\dfrac{\sin x+\cos x}{\sin x-\cos x}\), the value of \(f'(x)\) at \(x=0\) is ________________.
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For ratios, use the quotient rule and plug values after differentiating to avoid algebraic slips.
Updated On: Aug 26, 2025
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Correct Answer: -2
Solution and Explanation
Step 1: Let \(u=\sin x+\cos x\), \(v=\sin x-\cos x\). Then
\(f'=\dfrac{u'v-uv'}{v^2}\).
Step 2: At \(x=0\): \(u=1,\ v=-1,\ u'=1,\ v'=1\). Hence \[ f'(0)=\frac{(1)(-1)-(1)(1)}{(-1)^2}=\frac{-2}{1}=\boxed{-2}. \]
Step 2: At \(x=0\): \(u=1,\ v=-1,\ u'=1,\ v'=1\). Hence \[ f'(0)=\frac{(1)(-1)-(1)(1)}{(-1)^2}=\frac{-2}{1}=\boxed{-2}. \]
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