Q.Find $f'(x)$ if $f(x) = (\sin x)^{\sin x}$ for all $0 < x < \pi$.

Concept: Implicit Differentiation — when the variable appears in both the base and the exponent, take logs to bring the exponent down, then differentiate implicitly.

Let $y = (\sin x)^{\sin x}$. Take the natural logarithm of both sides:

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

Simplify the last term: $\sin x \cdot \frac{\cos x}{\sin x} = \cos x$. So: …

When you see a function where the variable appears in both the base and the exponent — like $(\sin x)^{\sin x}$ — the standard power rule or exponential rule alone won't work. The power rule assumes a constant exponent; the exponential rule assumes a constant base. Here, both are moving.

The trick is to use logarithmic differentiation. Taking the natural logarithm transforms the exponent into a coefficient, letting you differentiate using the product rule. Then you solve for $f'(x)$ by multiplying back the original function.

Let’s walk through it.

1. Set up the equation. Write $y = f(x) = (\sin x)^{\sin x}$. Take the natural logarithm of both sides:

Using the logarithm power rule: $\log(a^b) = b \log a$, we get:

2. Differentiate implicitly with respect to $x$. On the left, the derivative of $\log y$ is $\frac{1}{y} \cdot y'$ (by the chain rule). On the right, we have a product: $\sin x$ times $\log(\sin x)$. Use the product rule:

The second term simplifies: $\sin x \cdot \frac{\cos x}{\sin x} = \cos x$.

So the derivative of the right side is:

Factor out $\cos x$:

Putting it together:

3. Solve for $y'$. Multiply both sides by $y$: …

Method: Logarithmic Differentiation for a Variable Base and Variable Exponent

Use this whenever a function has the form $y = [u(x)]^{v(x)}$ — both the base and the exponent depend on $x$, so neither the plain power rule (which needs a constant exponent) nor the plain exponential rule (which needs a constant base) applies on its own.

Steps

Step 1: Recognise the form

Check whether the variable appears in both the base and the exponent. If it does, you cannot differentiate directly — you need logarithmic differentiation first.

Step 2: Take logs of both sides

Write $y = [u(x)]^{v(x)}$ and take the natural log:

This converts the troublesome exponent into a coefficient, turning the right side into an ordinary product.

Step 3: Differentiate both sides implicitly with respect to $x$

The left side differentiates to $\dfrac{1}{y}\dfrac{dy}{dx}$ (chain rule). The right side is a product, so use the product rule:

Step 4: Solve for the derivative …

Common Mistakes

Mistake 1: Applying the plain power rule to $(\sin x)^{\sin x}$

Why it's wrong: the power rule $\frac{d}{dx}u^n = n\,u^{n-1}u'$ only works when the exponent $n$ is a constant. Here the exponent $\sin x$ itself varies with $x$, so this rule silently drops an entire term. Correct approach: since both base and exponent depend on $x$, take logs first and use logarithmic differentiation.

Mistake 2: Dropping a term when differentiating $\sin x\cdot\log(\sin x)$

Why it's wrong: this is a product of two $x$-dependent factors, so it needs the product rule — $\frac{d}{dx}\big[\sin x\log(\sin x)\big]=\cos x\log(\sin x)+\sin x\cdot\frac{\cos x}{\sin x}$. Students often differentiate only one factor and forget the other. Correct approach: write out both product-rule terms explicitly before simplifying. …