Q.Find $\frac{dy}{dx}$ in the following: $\frac{e^x}{\sin x}$

The key idea is Implicit Differentiation — but here the function is already explicit, so we simply differentiate using the quotient rule.

Let $y = \frac{e^x}{\sin x}$.

Step 1: Apply the quotient rule:

$\frac{dy}{dx} = \frac{(\sin x)(e^x)' - (e^x)(\sin x)'}{\sin^2 x}$.

Step 2: Differentiate:

$(e^x)' = e^x$, $(\sin x)' = \cos x$.

Step 3: Substitute:

$\frac{dy}{dx} = \frac{e^x \sin x - e^x \cos x}{\sin^2 x} = e^x \cdot \frac{\sin x - \cos x}{\sin^2 x}$.

The problem asks for $\frac{dy}{dx}$ of $y = \frac{e^x}{\sin x}$. This is a straightforward derivative of a quotient of two functions: $e^x$ in the numerator and $\sin x$ in the denominator. The natural tool here is the quotient rule, but we could also rewrite the function as $e^x \cdot \csc x$ and use the product rule — both lead to the same result.

Let’s work through it step by step.

1. Identify the functions.

   Let $u = e^x$ and $v = \sin x$. Then $y = \frac{u}{v}$.

2. Recall the quotient rule.

   For $y = \frac{u}{v}$,

This formula comes from the limit definition of the derivative, but the intuition is: the rate of change of a ratio depends on how fast the top and bottom change relative to each other.

3. Compute the derivatives.

   • $\frac{du}{dx} = \frac{d}{dx} e^x = e^x$ (the exponential function is its own derivative).
   • $\frac{dv}{dx} = \frac{d}{dx} \sin x = \cos x$.

4. Plug into the quotient rule.

5. Simplify the numerator. Factor out $e^x$:

This is a perfectly acceptable final form. However, we can also write it in terms of cosecant and cotangent if desired:

Method: The Quotient Rule for Ratios of Standard Functions

Use this method whenever $y$ is written as one differentiable function divided by another, and there's no obvious simplification that removes the division.

Steps

Step 1: Identify the numerator and denominator

Write $y = \dfrac{u}{v}$ and clearly name $u$ and $v$ (here $u=e^x$, $v=\sin x$).

Step 2: Recall and apply the quotient rule formula

The order matters: it's "bottom times derivative of top, minus top times derivative of bottom," all over the bottom squared.

Step 3: Differentiate $u$ and $v$ separately using standard derivative rules

Compute $\dfrac{du}{dx}$ and $\dfrac{dv}{dx}$ independently before combining — this keeps errors isolated and easy to check.

Step 4: Substitute into the formula and simplify

Plug the two derivatives into the quotient-rule formula, then factor out any common terms (such as $e^x$) to present the answer in its simplest form.

Common Mistakes

Mistake 1: Swapping the order of subtraction in the quotient rule

Why it's wrong: writing $u\,dv - v\,du$ instead of $v\,du - u\,dv$ flips the sign of the whole result. Correct approach: always say it out loud as "bottom times d(top) minus top times d(bottom)" before writing the formula.

Mistake 2: Forgetting to square the denominator

Why it's wrong: the quotient rule's denominator is $v^2$, not $v$ — omitting the square gives a dimensionally wrong derivative. Correct approach: write the full formula with $v^2$ first, then fill in the pieces.

Mistake 3: Not factoring out the common term at the end

Why it's wrong: leaving the answer as $\dfrac{e^x\sin x - e^x\cos x}{\sin^2 x}$ instead of $\dfrac{e^x(\sin x - \cos x)}{\sin^2 x}$ isn't incorrect, but it can cause mismatches when comparing to a marking scheme or textbook answer key that expects the factored form. Correct approach: always check for a common factor in the numerator before finalizing.