Q.Differentiate $x^{\sin x}$, $x > 0$ w.r.t. $x$.

Concept: Implicit Differentiation (taking logs to handle a variable exponent).

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

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

Multiply through by $y = x^{\sin x}$: …

The function $x^{\sin x}$ is not a simple power function (where the exponent is constant) nor a simple exponential (where the base is constant). The variable appears in both the base and the exponent. To handle this, we use a technique called logarithmic differentiation — which is really just implicit differentiation in disguise.

The core idea: take the natural logarithm of both sides, use log properties to bring the exponent down, then differentiate implicitly. This converts the problem into a product rule inside a chain rule, which is much easier to manage.

1. Set up the equation Let $y = x^{\sin x}$, with $x > 0$ (so $\log x$ is defined). Take the natural log of both sides:

2. Differentiate implicitly with respect to $x$ On the left, $\frac{d}{dx}(\log y) = \frac{1}{y} \cdot \frac{dy}{dx}$ (by the chain rule). On the right, $\sin x \cdot \log x$ is a product, so we use the product rule:

Putting it together:

3. Solve for $\frac{dy}{dx}$ Multiply both sides by $y$:

Now substitute back $y = x^{\sin x}$:

$\frac{dy}{dx} = x^{\sin x} \left( \cos x \cdot \log x + \frac{\sin x}{x} \right)$ …

Method: Logarithmic Differentiation When Both the Base and the Exponent Contain the Variable

Use this method whenever a function has the variable $x$ in both the base and the exponent, such as $x^{\sin x}$, $(\sin x)^x$, or $x^x$ — neither the power rule (constant exponent) nor the exponential rule (constant base) applies directly.

Steps

Step 1: Set $y$ equal to the given function and take the natural log of both sides

Step 2: Differentiate both sides with respect to $x$

The left side gives $\dfrac{1}{y}\dfrac{dy}{dx}$ by the chain rule. The right side is now a product of the exponent and $\log(\text{base})$, so apply the product rule: …

Common Mistakes

Mistake 1: Applying the power rule because the base looks like "$x$ to a power"

Why it's wrong: The exponent $\sin x$ is not a constant, so the power rule ($n x^{n-1}$) does not apply — writing $\sin x \cdot x^{\sin x - 1}$ incorrectly assumes the exponent stays fixed while only the base varies. Correct approach: whenever the exponent itself contains $x$, take logs first instead of reaching for the power rule.

Mistake 2: Applying the exponential rule because the exponent looks like "something to the $x$" …