Q.Find $\frac{dy}{dx}$ in the following: $x^{\cos x} + \frac{x^2+1}{x^2-1}$

Concept: Implicit Differentiation — but here we only need standard differentiation rules (product, logarithmic, and quotient).

Let $y = x^{\cos x} + \frac{x^2+1}{x^2-1}$. Differentiate term by term.

Step 1: For $u = x^{\cos x}$, take $\log u = \cos x \cdot \log x$. Differentiate:

So

Step 2: For $v = \frac{x^2+1}{x^2-1}$, use quotient rule: …

The problem asks for $\frac{dy}{dx}$ where

This is a sum of two very different-looking pieces. The second piece is a straightforward rational function — quotient rule territory. The first piece, $x^{\cos x}$, is trickier: the variable $x$ appears in both the base and the exponent. That’s a classic signal for logarithmic differentiation.

Let’s break it down.

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1. Differentiate $x^{\cos x}$ using logarithmic differentiation

Why can’t we just use the power rule or the exponential rule directly? The power rule ($\frac{d}{dx} x^n = n x^{n-1}$) assumes the exponent is constant. The exponential rule ($\frac{d}{dx} a^x = a^x \log a$) assumes the base is constant. Here, both change with $x$, so neither applies directly.

The trick: take the natural log of both sides, use log properties to bring the exponent down, then differentiate implicitly.

Let $u = x^{\cos x}$. Then

Now differentiate both sides with respect to $x$. On the left, by the chain rule:

On the right, use the product rule:

So

Multiply through by $u = x^{\cos x}$:

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2. Differentiate $\frac{x^2+1}{x^2-1}$ using the quotient rule

Let $v = \frac{x^2+1}{x^2-1}$. The quotient rule says:

$\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f'g - fg'}{g^2}.$ …

Method: Splitting a Sum into the Right Differentiation Rule for Each Term

When a derivative problem gives you a sum of unlike pieces, don't reach for one formula — identify the right rule for each piece separately, differentiate them independently, then add the results (by the sum rule).

Steps

Step 1: Scan each term and classify its type

Look at every term added or subtracted in the expression. A term of the form $[\text{base}(x)]^{[\text{exponent}(x)]}$ — where both the base and the exponent depend on $x$ — cannot be handled by the power rule ($x^n$) or the exponential rule ($a^x$) alone; it needs logarithmic differentiation. A term that is one expression divided by another needs the quotient rule.

Step 2: Logarithmic differentiation for a variable-base, variable-exponent term

For $u = f(x)^{g(x)}$:

Differentiate both sides with respect to $x$ (product rule on the right, chain rule on the left):

then multiply through by $u$ to isolate $\dfrac{du}{dx}$.

Step 3: Quotient rule for a fraction of two functions

For $v = \dfrac{p(x)}{q(x)}$:

$\frac{dv}{dx} = \frac{p'(x)q(x) - p(x)q'(x)}{[q(x)]^2}$ …

Common Mistakes

Mistake 1: Applying the ordinary power or exponential rule to $x^{\cos x}$

Why it's wrong: the power rule needs a constant exponent, and the exponential rule needs a constant base — here both $x$ (the base) and $\cos x$ (the exponent) depend on $x$, so neither applies. A student who writes $\frac{d}{dx}x^{\cos x} = \cos x \cdot x^{\cos x -1}$ (treating $\cos x$ as if it were fixed) gets a completely wrong expression. Correct approach: take $\log$ of the term first, differentiate implicitly, then multiply back by $x^{\cos x}$.

Mistake 2: Losing the sign in the quotient rule …