Question

If $y = \frac{(a \cos x + b \sin x + c)}{\sin x}$ then $\frac{dy}{dx} = $

• A. $-a \csc^{2} x - c \csc x \cot x$
• B. $-a \csc^{2} x$
• C. $-a \csc^{2} x + b \sec^{2} x + c \csc x \cot x$
• D. $a \csc^{2} x - c \csc x \cot x$

Hint:

Split the fraction! It's much faster than using the Quotient Rule for simple denominators.

Answer 1

The Correct Option is A

Step 1: Concept
Simplify the expression by dividing each term by $\sin x$ before differentiating.

Step 2: Meaning
$y = a \cot x + b + c \csc x$.

Step 3: Analysis
Differentiate with respect to $x$: $d/dx(a \cot x) = -a \csc^{2} x$, $d/dx(b) = 0$, and $d/dx(c \csc x) = -c \csc x \cot x$.

Step 4: Conclusion
Summing the derivatives: $dy/dx = -a \csc^{2} x - c \csc x \cot x$.
Final Answer: (A)