What is the result when differentiate f(x)=sin(x^3+square rootx*ln(3x+sin 2x))?

Since we notice that the given functin is a product of
composed functions, we'll differentiate with respect to x, using product rule and chain
rule.

We'll apply first the product
rule:

f'(x)=[sin(x^3 + sqrtx)]'*ln(3x+sin 2x) +
[sin(x^3+square rootx)]*[ln(3x+sin 2x)]'

Now, w'ell apply
chain rule:

f'(x) =
[cos(x^3 + sqrtx)]*(x^3 + sqrtx)'*ln(3x+sin 2x) + [sin(x^3+square rootx)]*[(3 + 2cos
2x)/(3x+sin 2x)]

f'(x) = (3x^2 + 1/2sqrtx)*[cos(x^3 +
sqrtx)]*ln(3x+sin 2x) + [sin(x^3+square rootx)]*[(3 + 2cos 2x)/(3x+sin
2x)]

The result of differentiation
is:

f'(x) = (3x^2 + 1/2sqrtx)*[cos(x^3 +
sqrtx)]*ln(3x+sin 2x) + [sin(x^3+square rootx)]*[(3 + 2cos 2x)/(3x+sin
2x)]