Q.Differentiate $\sin^2 x$ w.r.t. $e^{\cos x}$.

The key idea is the Chain Rule applied to parametric differentiation — we treat $\sin^2 x$ as a function of $x$, and $e^{\cos x}$ as another function of $x$, then use:

Step 1: Differentiate $\sin^2 x$ with respect to $x$:

Step 2: Differentiate $e^{\cos x}$ with respect to $x$: …

The question asks us to differentiate one function with respect to another — not the usual $dy/dx$ where both are functions of $x$. This is a classic "parametric differentiation" problem. Think of it this way: both $\sin^2 x$ and $e^{\cos x}$ are expressed in terms of a common parameter $x$. So we can find the derivative of the first with respect to the second by taking the ratio of their individual derivatives with respect to $x$.

The chain rule is the backbone here. If $y = \sin^2 x$ and $u = e^{\cos x}$, then:

provided $du/dx \neq 0$. This is simply the derivative of $y$ with respect to $x$ divided by the derivative of $u$ with respect to $x$.

Let's work through it step by step.

1. Set up the functions.

   Let $y = \sin^2 x$ and $u = e^{\cos x}$. We need $\frac{dy}{du}$.

2. Differentiate $y = \sin^2 x$ with respect to $x$.

   This is a composition: $(\sin x)^2$. Using the chain rule:

Alternatively, you can write $2 \sin x \cos x = \sin 2x$, but we'll keep it as $2 \sin x \cos x$ for now.

3. Differentiate $u = e^{\cos x}$ with respect to $x$. Again, chain rule: derivative of $e^{\text{something}}$ is $e^{\text{something}}$ times the derivative of that something.

4. Form the ratio.

5. Simplify. …

Method: Differentiating One Function with Respect to Another (via a Common Variable)

Use this whenever you're asked to differentiate one expression "with respect to" another expression, rather than with respect to $x$ — both expressions are really functions of the same underlying variable $x$.

Steps

Step 1: Recognise the form

"Differentiate $f(x)$ with respect to $g(x)$" means find $\dfrac{df}{dg}$, not $\dfrac{df}{dx}$. There is no direct rule for this — both $f$ and $g$ must first be related back to their common variable $x$.

Step 2: Rename and connect via the chain rule

Let $y = f(x)$ and $u = g(x)$. By the chain rule,

Rearranging (valid wherever $\dfrac{du}{dx}\neq 0$):

Step 3: Differentiate each piece separately with respect to $x$

Find $\dfrac{dy}{dx}$ and $\dfrac{du}{dx}$ independently, using the ordinary differentiation rules (chain rule, product rule, etc.) — treat this as two separate, standard differentiation problems. …

Common Mistakes

Mistake 1: Trying to differentiate $\sin^2 x$ "directly" with respect to $e^{\cos x}$

Why it's wrong: there is no standard differentiation rule for one arbitrary expression with respect to another — both must first be related back to the common variable $x$ via $\frac{dy}{du}=\frac{dy/dx}{du/dx}$. Correct approach: introduce $y=\sin^2x$, $u=e^{\cos x}$, differentiate each with respect to $x$ separately, then take the ratio.

Mistake 2: Dropping the negative sign when differentiating $e^{\cos x}$

Why it's wrong: $\frac{d}{dx}\cos x=-\sin x$, so $\frac{d}{dx}e^{\cos x}=-e^{\cos x}\sin x$; forgetting the minus sign flips the sign of the entire final answer. Correct approach: apply the chain rule to the exponent carefully, keeping the $-\sin x$ factor. …