Find nth differentiation of following functions: sin 2x cos3x

To find the \(n\)-th derivative of \(f(x) = \sin(2x) \cos(3x)\), we can simplify the expression using a trigonometric identity. First, let's apply the product-to-sum formula:

\[
\sin(A)\cos(B) = \frac{1}{2} [\sin(A+B) + \sin(A-B)]
\]

For \(A = 2x\) and \(B = 3x\), this gives us:

\[
\sin(2x)\cos(3x) = \frac{1}{2} [\sin(5x) + \sin(-x)]
\]
Since \(\sin(-x) = -\sin(x)\), we have:

\[
f(x) = \frac{1}{2} [\sin(5x) - \sin(x)]
\]

Now, we can differentiate this function \(n\)-times.

### Differentiating \(f(x)\):

1. First derivative:
\[
f'(x) = \frac{1}{2} [5 \cos(5x) - \cos(x)]
\]

2. Second derivative:
\[
f''(x) = \frac{1}{2} [-25 \sin(5x) + \sin(x)]
\]

3. Third derivative:
\[
f^{(3)}(x) = \frac{1}{2} [-125 \cos(5x) - \cos(x)]
\]

### General Pattern:
The \(n\)-th derivative of \(\sin(kx)\) alternates between sine and cosine, and the coefficient changes according to powers of \(k\). For \(\sin(5x)\), the derivatives alternate between \(\sin(5x)\) and \(\cos(5x)\), with the coefficients being powers of 5. Similarly, for \(\sin(x)\), the coefficients are powers of 1.

The \(n\)-th derivative of the function is:

\[
f^{(n)}(x) = \frac{1}{2} [5^n \sin(5x + n\frac{\pi}{2}) - \sin(x + n\frac{\pi}{2})]
\]

This expression accounts for the alternating sine and cosine forms and the increasing powers of 5.

Would you like a deeper explanation or details on a specific part?

Here are some related questions:

1. How do trigonometric identities simplify differentiation?
2. What is the general rule for the \(n\)-th derivative of sine and cosine?
3. How do phase shifts affect trigonometric derivatives?
4. Can the product rule be used instead of identities here?
5. What other functions benefit from product-to-sum identities?

**Tip:** The alternating sine and cosine forms in higher derivatives follow a cycle every four derivatives, due to their periodic nature.

Find nth differentiation of following functions: sin 2x cos 3x

Explore the process of finding the nth derivative of sin(2x)cos(3x) using product-to-sum identities and higher-order differentiation. This detailed solution covers trigonometric identities, derivative patterns, and power rules. Suitable for advanced high school and undergraduate calculus students.

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