If y = cos 3x, find $\frac{\delta y}{\delta x}$

Analysis

The question asks for the derivative of the trigonometric function $y = \cos 3x$ with respect to $x$. This is a classic calculus problem involving the Chain Rule.

Concept: The Chain Rule

When you have a composite function like $y = \cos(u)$, where $u$ is a function of $x$ (in this case, $u = 3x$), you use the chain rule:

1. Derivative of $\cos(u)$: The derivative of $\cos(u)$ with respect to $u$ is $-\sin(u)$.
2. Derivative of $3x$: The derivative of $3x$ with respect to $x$ is $3$.

Step-by-Step Solution

1. Identify the components: Let $u = 3x$. Then $y = \cos(u)$.

2. Differentiate each part: $\frac{dy}{du} = -\sin(u)$ $\frac{du}{dx} = 3$

3. Apply the Chain Rule: $\frac{dy}{dx} = (-\sin(u)) \cdot (3)$

4. Substitute $u = 3x$ back into the equation: $\frac{dy}{dx} = -3 \sin(3x)$

Conclusion

Comparing our result to the given options, we find that our calculated derivative matches option D.

Exam Tip: Always remember the "negative" sign when differentiating cosine functions. A common mistake students make is forgetting that the derivative of $\cos(x)$ is $-\sin(x)$, not $+\sin(x)$. Also, don't forget to multiply by the derivative of the "inside" function (the 3 in $3x$).