Q.Find $\frac{dy}{dx}$ in the following: $\sin (\log x)$

Concept: Chain Rule — differentiate the outer function first, then multiply by the derivative of the inner function.

Let $y = \sin(\log x)$.

The outer function is $\sin u$, where $u = \log x$.

Step 1: $\frac{dy}{du} = \cos u = \cos(\log x)$.

Step 2: $\frac{du}{dx} = \frac{1}{x}$ (since $\log x$ is natural log). …

The key idea here is the Chain Rule. Whenever you have a function of a function — like $\sin$ of $\log x$ — you cannot differentiate it in one step. The sine function “sees” $\log x$ as its input, not $x$ directly. So you first differentiate the outer function (sine) with respect to its own input, and then multiply by the derivative of that inner input ($\log x$) with respect to $x$.

Let’s walk through it carefully.

1. Identify the outer and inner functions.

   We have $y = \sin(\log x)$.

   The outer function is $\sin(u)$, where $u = \log x$ is the inner function.

2. Differentiate the outer function with respect to its inner input.

   The derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$.

   So at this stage, we have $\frac{dy}{du} = \cos(u) = \cos(\log x)$.

3. Differentiate the inner function with respect to $x$.

   The inner function is $u = \log x$. Its derivative is $\frac{du}{dx} = \frac{1}{x}$.

   Note

   Here $\log x$ means the natural logarithm (base $e$), as is standard in calculus. Its derivative is $1/x$.

4. Apply the Chain Rule.

   The Chain Rule says: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

   Substituting what we have: …

Method: The Chain Rule for a Composite Function

Whenever a function is "wrapped inside" another function — $f(g(x))$ — differentiate the outer function first (with respect to its own argument), then multiply by the derivative of the inner function.

Steps

Step 1: Identify the outer function and the inner function

Write $y=f(u)$ where $u=g(x)$ is everything "inside" the outermost operation.

Step 2: Differentiate the outer function with respect to $u$

Use the standard derivative rule for whatever the outer function is (log, power, trig, exponential, ...), keeping $u$ untouched.

Step 3: Differentiate the inner function $u$ with respect to $x$

Step 4: Multiply the two results

$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}.$ …

Common Mistakes

Mistake 1: Writing $\cos(\log x)$ as the final answer and forgetting the $\frac{1}{x}$ factor.

Why it's wrong: $\frac{1}{x}$ is exactly the derivative of the inner function $\log x$, required by the chain rule — omitting it would be correct only if the argument were $x$ itself, not $\log x$. Correct approach: always multiply by the derivative of whatever sits inside the outer function.

Mistake 2: Confusing $\sin(\log x)$ with $\log(\sin x)$. …