Q.Find $\frac{dy}{dx}$ in the following: $e^{6x} \cos 3x$

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

We have $y = e^{6x} \cos 3x$. This is a product, so first use the Product Rule:

Now apply the Chain Rule to each derivative:

• $\frac{d}{dx}(\cos 3x) = -\sin 3x \cdot 3 = -3\sin 3x$
• $\frac{d}{dx}(e^{6x}) = e^{6x} \cdot 6 = 6e^{6x}$

Substitute back: …

We are given $y = e^{6x} \cos 3x$. This is a product of two functions: $u = e^{6x}$ and $v = \cos 3x$. To differentiate a product, we use the Product Rule: $\frac{d}{dx}(uv) = u'v + uv'$.

But each of these functions is itself a composition — $e^{6x}$ is $e$ raised to $6x$, and $\cos 3x$ is $\cos$ of $3x$. So each requires the Chain Rule as well. Let’s break it down step by step.

1. Differentiate $u = e^{6x}$. The derivative of $e^{\text{something}}$ is $e^{\text{something}}$ times the derivative of that something. Here, the "something" is $6x$, whose derivative is $6$. So:

2. Differentiate $v = \cos 3x$. The derivative of $\cos(\text{something})$ is $-\sin(\text{something})$ times the derivative of that something. Here, the "something" is $3x$, whose derivative is $3$. So:

3. Apply the Product Rule. The Product Rule says:

Substitute $u$, $u'$, $v$, and $v'$:

4. Simplify. Factor out the common factor $e^{6x}$: …

Method: The Product Rule (with Chain Rule on Each Factor)

When two functions of $x$ are multiplied together, neither the sum rule nor differentiating each factor separately and multiplying works — the product rule is required.

Steps

Step 1: Identify the two factors $u(x)$ and $v(x)$ being multiplied

Step 2: Differentiate each factor separately

If either factor is itself composite, apply the chain rule to it individually at this stage.

Step 3: Combine using the product rule

$\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}.$ …

Common Mistakes

Mistake 1: Applying the chain rule only partially — e.g. writing $\frac{d}{dx}\cos3x=-\sin3x$ without the factor of $3$.

Why it's wrong: both factors here are themselves composite ($e^{6x}$ and $\cos3x$), so each needs its own inner-derivative multiplied in ($6$ for the exponential, $3$ for the cosine) — dropping either changes the answer's magnitude. Correct approach: differentiate each factor as a full mini chain-rule problem before combining with the product rule.

Mistake 2: Losing the minus sign from $\frac{d}{dx}\cos3x=-3\sin3x$ when combining terms. …