Q.Differentiate the function $\cos(\log x + e^x)$ with respect to $x$, where $x > 0$.

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

Step 1: Let $u = \log x + e^x$. Then the function is $\cos u$.

Step 2:

$\frac{d}{dx} \cos u = -\sin u \cdot \frac{du}{dx}$.

Step 3:

$\frac{du}{dx} = \frac{1}{x} + e^x$.

Step 4: Substitute back $u = \log x + e^x$: …

The key idea here is the Chain Rule. When you have a function of a function — like $\cos$ of something that itself depends on $x$ — you differentiate the outer function first, then multiply by the derivative of the inner function. Think of it as peeling an onion: the outermost layer is $\cos$, then inside is $(\log x + e^x)$.

Let’s walk through it step by step.

1. Identify the outer and inner functions.

   The given function is $\cos(\log x + e^x)$.

   • Outer function: $\cos(u)$, where $u$ is a placeholder.
   • Inner function: $u = \log x + e^x$.

2. Differentiate the outer function with respect to its argument.

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

   So, $\frac{d}{du}[\cos(u)] = -\sin(u)$.

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

   $u = \log x + e^x$.

   • Derivative of $\log x$ is $\frac{1}{x}$ (since $x > 0$, this is well-defined).
   • Derivative of $e^x$ is $e^x$. Hence, $\frac{du}{dx} = \frac{1}{x} + e^x$.

4. Apply the Chain Rule.

   The Chain Rule says:

Substitute what we have: …

Method: Differentiating a Composite Function by the Chain Rule

This method applies whenever the function you're asked to differentiate is one function "wrapped around" another — here, cosine wrapped around a sum of $\log x$ and $e^x$.

Steps

Step 1: Identify the outer function and the inner function.

Write the given expression as $f(u)$ where $u$ is everything inside the outer function. For $\cos(\log x + e^x)$, the outer function is $f(u) = \cos u$ and the inner function is $u = \log x + e^x$.

Step 2: Differentiate the outer function with respect to $u$, keeping $u$ unevaluated.

Do not substitute the inner expression yet — this step only produces the "shape" of the outer derivative.

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

Differentiate each piece of $u$ separately using standard rules — the derivative of $\log x$ is $\frac{1}{x}$, and the derivative of $e^x$ is $e^x$: …

Common Mistakes

Mistake 1: Dropping the negative sign from the derivative of cosine.

Why it's wrong: $\frac{d}{du}\cos u = -\sin u$, not $\sin u$ — the minus sign is part of the rule, not optional. Correct approach: always write $-\sin u$ the moment you differentiate a cosine, before multiplying by anything else.

Mistake 2: Differentiating only part of the inner function. …