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

Concept: Derivative Evaluation — Use the chain rule with an exponential base $a$.

Let $y = 2^{\cos^2 x}$.

Take $\log$ both sides: $\log y = \cos^2 x \cdot \log 2$.

Differentiate: $\frac{1}{y} \frac{dy}{dx} = \log 2 \cdot 2\cos x \cdot (-\sin x) = -\log 2 \cdot \sin 2x$. …

Concept and Intuition

When you see a function like $a^{f(x)}$ — a constant base raised to a variable exponent — the standard approach is to use the exponential form. Why? Because the derivative of $a^x$ is $a^x \log a$, but that rule only works when the exponent is exactly $x$. Here, the exponent is $\cos^2 x$, a function of $x$, so we need the chain rule.

The cleanest way is to rewrite $2^{\cos^2 x}$ as $e^{\cos^2 x \cdot \log 2}$. This turns the problem into differentiating $e^{u(x)}$, where $u(x) = \cos^2 x \cdot \log 2$. The derivative of $e^{u}$ is $e^{u} \cdot u'$, and then we just need $u'$.

Step-by-Step Solution

1. Rewrite in exponential form Let $y = 2^{\cos^2 x}$. Then

2. Differentiate using the chain rule The derivative of $e^{u}$ is $e^{u} \cdot \frac{du}{dx}$. Here $u = \cos^2 x \cdot \log 2$, so

3. Factor out the constant $\log 2$ is a constant, so

4. Differentiate $\cos^2 x$ …

Method: Differentiating $a^{f(x)}$ — Constant Base, Variable Exponent

Use this method whenever you must differentiate an expression of the form $a^{f(x)}$, where $a$ is a fixed positive constant (not $e$) and the exponent $f(x)$ is itself a function of $x$.

Steps

Step 1: Recognise the exponential form and convert the base

The direct formula $\frac{d}{dx}(a^x) = a^x \ln a$ only applies when the exponent is exactly $x$. When the exponent is a function $f(x)$, rewrite $a^{f(x)}$ in base $e$:

(equivalently, take logarithms of both sides — this is logarithmic differentiation applied to a single exponential term).

Step 2: Differentiate the exponential using the chain rule

Since $\ln a$ is a constant, $\dfrac{du}{dx} = \ln a \cdot f'(x)$.

Step 3: Differentiate the inner function $f(x)$ on its own …

Common Mistakes

Mistake 1: Forgetting the $\ln a$ factor entirely

Why it's wrong: students often treat $2^{\cos^2 x}$ the same way as $e^{\cos^2 x}$, whose derivative needs no extra constant. But the base here is $2$, not $e$, so a factor of $\ln 2$ is unavoidable in the derivative — omitting it gives an answer that is off by a constant multiple everywhere. Correct approach: always convert $a^{f(x)}$ to $e^{f(x)\ln a}$ (or explicitly recall $\frac{d}{dx}a^{u}=a^u \ln a \cdot u'$) before differentiating.

Mistake 2: Differentiating only the outer exponential and forgetting the inner chain rule on $\cos^2 x$ …