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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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$\frac{\frac{d}{dt}\left(at\right)\left(1+bt^2\right)^3-at\frac{d}{dt}\left(\left(1+bt^2\right)^3\right)}{\left(\left(1+bt^2\right)^3\right)^2}$
Learn how to solve quotient rule of differentiation problems step by step online. Find the derivative d/dt((at)/((1+bt^2)^3)). 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}}. Simplify \left(\left(1+bt^2\right)^3\right)^2 using the power of a power property: \left(a^m\right)^n=a^{m\cdot n}. In the expression, m equals 3 and n equals 2. The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function. The derivative of the linear function is equal to 1.