Quotient Rule

The derivative of a function `f \left( x \right) = \frac{u \left( x \right)}{v \left( x \right)}` is $$f' \left( x \right) = \frac{u' \left( x \right) v \left( x \right) - u \left( x \right) v' \left( x \right)}{v \left( x \right)^{2}}$$


Let `f \left( x \right) = \frac{x^3}{\ln \left( x \right)}`. It is the quotient of 2 functions: `u \left( x \right) = x^3` and `v \left( x \right) = \ln \left( x \right)`.

The function `u \left( x \right) = x^{3}` is a power function `x^{n}` with `n=3`. Use the power rule $$ \begin{align*} u' \left( x \right) &= 3 x ^{3-1} \\ &= 3 x^{2} \end{align*} $$

The function `v \left( x \right) = \ln \left( x \right)` is the logarithmic function. It's derivative is know: $$ \begin{align*} v' \left( x \right) = \frac{1}{x} \end{align*} $$

The derivative of `f \left( x \right)` is $$ \begin{align*} f' \left( x \right) &= \frac{u' \left( x \right) v \left( x \right) - u \left( x \right) v' \left( x \right)}{v \left( x \right)^{2}} \\ &= \frac{3x^2 \ln \left( x \right) - x^{3} \frac{1}{x}}{\ln \left( x \right)^{2}} \\ &= \frac{3x^2 \ln \left( x \right) - x^{2}}{\ln \left( x \right)^{2}} \\ \end{align*} $$


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