Limits, Continuity & Differentiability
Derivatives of trigonometric functions
Grade 12
Question:
<p>If <i>f</i>(<i>x</i>) = \(\frac{\cos x}{\sin x}\), then <i>f</i>'(\(\frac{\pi}{4}\)) is equal to</p>
<p>(a) 2</p>
<p>(b) −2</p>
<p>(c) 0</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: The derivative of cot $x$ is −csc² $x$; evaluate at the given point using the known value csc(π/4) = √2.
<p><strong>Step 1:</strong> <i>f</i>(<i>x</i>) = $\frac{\cos x}{\sin x}$ = cot <i>x</i></p><p><strong>Step 2:</strong> <i>f</i>'(<i>x</i>) = −csc²<i>x</i></p><p><strong>Step 3:</strong> <i>f</i>'($\frac{\pi}{4}$) = −csc²($\frac{\pi}{4}$) = −($\sqrt{2}$)<sup>2</sup> = −2</p><p>∴ Answer is (b) −2</p>
Correct Answer: B