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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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$\frac{\sqrt{x-1}\frac{d}{dx}\left(x^8\cos\left(x\right)^3\right)-x^8\frac{d}{dx}\left(\sqrt{x-1}\right)\cos\left(x\right)^3}{\left(\sqrt{x-1}\right)^2}$
Learn how to solve problems step by step online. Find the derivative d/dx((x^8cos(x)^3)/((x-1)^1/2)). 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}}. Cancel exponents \frac{1}{2} and 2. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x^8 and g=\cos\left(x\right)^3. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}.