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Apply the quotient rule for differentiation, which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$
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$\frac{\left(2x+1\right)^5\left(3x+2\right)\frac{d}{dx}\left(x^2\left(x+2\right)^3\sin\left(x\right)^2\right)-x^2\left(x+2\right)^3\frac{d}{dx}\left(\left(2x+1\right)^5\left(3x+2\right)\right)\sin\left(x\right)^2}{\left(\left(2x+1\right)^5\left(3x+2\right)\right)^2}$
Learn how to solve problems step by step online. Find the derivative of (x^2(x+2)^3sin(x)^2)/((2x+1)^5(3x+2)). Apply the quotient rule for differentiation, which states that if f(x) and g(x) are functions and h(x) is the function defined by {\displaystyle h(x) = \frac{f(x)}{g(x)}}, where {g(x) \neq 0}, then {\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}. The power of a product is equal to the product of it's factors raised to the same power. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g'. Simplify the product -(\frac{d}{dx}\left(\left(2x+1\right)^5\right)\left(3x+2\right)+\left(2x+1\right)^5\frac{d}{dx}\left(3x+2\right)).