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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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$\frac{\frac{d}{dx}\left(x^3+x+1\right)x\left(2x-5\right)^3\left(x^2+2x+5\right)^2-\left(x^3+x+1\right)\frac{d}{dx}\left(x\left(2x-5\right)^3\left(x^2+2x+5\right)^2\right)}{\left(x\left(2x-5\right)^3\left(x^2+2x+5\right)^2\right)^2}$
Learn how to solve problems step by step online. Find the derivative of (x^3+x+1)/(x(2x-5)^3(x^2+2x+5)^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. Simplify the product -(x^3+x+1). Simplify the product -(x+1).