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

Concept: Chain Rule – differentiate the outer function, then multiply by the derivative of the inner function, repeating for each nested layer.

Let $y = \log (\cos e^x)$.

1. Outer function: $\log u$, derivative $\frac{1}{u}$.

   So $\frac{dy}{dx} = \frac{1}{\cos e^x} \cdot \frac{d}{dx}(\cos e^x)$.

2. Next layer: $\cos v$, derivative $-\sin v$.

   $\frac{d}{dx}(\cos e^x) = -\sin(e^x) \cdot \frac{d}{dx}(e^x)$.

3. Innermost: $e^x$, derivative $e^x$.

   Putting it together: …

We start with $y = \log(\cos(e^x))$. The key here is the chain rule: when a function is composed of several layers, you differentiate each layer in order, multiplying the results. Think of it like peeling an onion — start with the outermost function and work your way in.

The outermost function is $\log(\cdot)$ (natural logarithm). Its derivative is $\frac{1}{\text{inside}}$. Then we multiply by the derivative of the inside, which itself is a composition: $\cos(\cdot)$ followed by $e^x$. So we need two more chain rule applications.

Let’s go step by step.

1. Identify the layers.

   We have:

   • Outer: $\log(u)$, where $u = \cos(e^x)$
   • Middle: $\cos(v)$, where $v = e^x$
   • Inner: $e^x$

2. Differentiate the outermost layer.

   The derivative of $\log(u)$ with respect to $u$ is $\frac{1}{u}$. So:

3. Now differentiate the middle layer. $u = \cos(v)$, where $v = e^x$. The derivative of $\cos(v)$ with respect to $v$ is $-\sin(v)$. So:

4. Differentiate the innermost layer. $v = e^x$, and its derivative with respect to $x$ is simply $e^x$:

5. Apply the chain rule by multiplying all these derivatives. The chain rule says:

Substituting: …

Method: Peeling a Triple-Nested Composition with the Chain Rule

Use this whenever a function is built from three (or more) layers nested inside each other, such as $\log(\cos(e^x))$.

Steps

Step 1: Identify every layer, outside to inside

List the layers explicitly: outer $\log(u)$ where $u=\cos(e^x)$; middle $\cos(v)$ where $v=e^x$; inner $v = e^x$. Writing this out prevents skipping a layer.

Step 2: Differentiate the outermost layer with respect to its own argument

Leave the inner content ($\cos(e^x)$) untouched at this stage — just note its derivative is needed next.

Step 3: Differentiate the middle layer with respect to its own argument

again leaving the innermost content ($e^x$) as-is. …

Common Mistakes

Mistake 1: Stopping after differentiating only the outermost layer

Why it's wrong: writing $\dfrac{dy}{dx} = \dfrac{1}{\cos(e^x)}$ and stopping ignores that $\cos(e^x)$ is itself a composite function that still needs to be differentiated. Correct approach: explicitly count the layers before starting, and don't declare the derivative finished until every layer has contributed a factor.

Mistake 2: Dropping the minus sign from the derivative of cosine

Why it's wrong: $\dfrac{d}{dv}\cos v = -\sin v$; forgetting the minus sign flips the final answer's sign. Correct approach: write the standard derivative of $\cos$ with its sign every time, rather than relying on memory mid-calculation. …