Differentiate $\frac{2x}{\sin x}$ with respect to x.

1. Analyze the Question

We are asked to find the derivative of the function $y = \frac{2x}{\sin x}$ with respect to $x$. Since this is a quotient of two functions, we must use the Quotient Rule.

2. Explain the Concept

The Quotient Rule states that if you have a function $y = \frac{u}{v}$, its derivative is given by:

In this problem:

• $u = 2x \implies \frac{du}{dx} = 2$
• $v = \sin x \implies \frac{dv}{dx} = \cos x$

3. Show the Solution

Substitute these values into the Quotient Rule formula:

Simplify the expression:

Now, split the fraction into two parts:

Simplify each term:

• For the first term: $\frac{2\sin x}{\sin^2 x} = \frac{2}{\sin x} = 2\csc x$
• For the second term: $\frac{2x\cos x}{\sin^2 x} = 2x \cdot \left(\frac{\cos x}{\sin x}\right) \cdot \left(\frac{1}{\sin x}\right) = 2x \cot x \csc x$

So, the derivative is:

Factor out $2\csc x$:

4. Identify the Answer

Comparing our result to the given options, we find that it matches option D.

5. Exam Tip

When dealing with trigonometric functions in calculus, always remember your basic identities: $\frac{1}{\sin x} = \csc x$ and $\frac{\cos x}{\sin x} = \cot x$. Recognizing these early makes simplifying your final answer much easier and helps you match it to the JAMB options quickly!