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X^3sin(2x+3) nth derivative?

Finding the nth Derivative of f(x) = x^3sin(2x+3)

Step 1: Write the Function in Product Form

The first step is to write the function in product form using the product rule of differentiation:

f(x) = x^3sin(2x+3) = x^3 * sin(2x+3)

Step 2: Find the Derivative of the Function

Using the product rule of differentiation, we get:

f'(x) = 3x^2sin(2x+3) + x^3cos(2x+3) * 2

f'(x) = 3x^2sin(2x+3) + 2x^3cos(2x+3)

Step 3: Find the Second Derivative of the Function

Using the product rule of differentiation again, we get:

f''(x) = 6xsin(2x+3) + 6x^2cos(2x+3) - 4x^3sin(2x+3)

f''(x) = 6x(sin(2x+3) + xcos(2x+3)) - 4x^3sin(2x+3)

Step 4: Find the Third Derivative of the Function

Using the product rule of differentiation again, we get:

f'''(x) = 6sin(2x+3) + 18xcos(2x+3) - 12x^2sin(2x+3) - 12x^2cos(2x+3)

f'''(x) = (6-12x^2)sin(2x+3) + 18xcos(2x+3) - 12x^2cos(2x+3)

Step 5: Find the nth Derivative of the Function

Using the product rule of differentiation again, we can find the nth derivative of the function by following the same process as above. The nth derivative of the function will have the form:

f^(n)(x) = (ax^3+b)sin(2x+3) + (cx^3+dx^2+ex+f)cos(2x+3)

where a, b, c, d, e, and f are constants that depend on the value of n.

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