Find the derivatives of y = sin 4x

Analysis

To find the derivative of $y = \sin 4x$, we need to apply the Chain Rule of differentiation. The Chain Rule is used when we have a composite function (a function inside another function).

Concept: The Chain Rule

The Chain Rule states that if you have a function $y = f(g(x))$, then the derivative is:

In this specific problem:

1. The "outer" function is $\sin(u)$, where $u = 4x$. The derivative of $\sin(u)$ is $\cos(u)$.
2. The "inner" function is $u = 4x$. The derivative of $4x$ with respect to $x$ is $4$.

Step-by-Step Solution

1. Identify the components: Let $u = 4x$. Then $y = \sin(u)$.

2. Differentiate the outer function: $\frac{dy}{du} = \cos(u)$

3. Differentiate the inner function: $\frac{du}{dx} = 4$

4. Multiply them together (Chain Rule): $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ $\frac{dy}{dx} = \cos(u) \cdot 4$

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

Conclusion

Comparing our result to the given options, we can see that $4 \cos 4x$ matches option D.

Exam Tip

Whenever you differentiate a trigonometric function like $\sin(ax)$ or $\cos(ax)$, always remember to multiply by the derivative of the angle ($a$).

• Derivative of $\sin(ax) = a \cos(ax)$
• Derivative of $\cos(ax) = -a \sin(ax)$ This quick mental check will save you time in the exam hall!