Quotient rule (film)

Introduction: If a function $y(x)$ is the quotient of two functions $u(x)$ and $v(x)$, then we can use the quotient rule to determine the derivative of $y(x)$.

Rule: Let $y(x) = \dfrac{u(x)}{v(x)}$. Then:

$$y'(x) = \dfrac{u'(x)v(x) - u(x)v'(x)}{\big(v(x)\big)^2}.$$

Example: Consider the function $y(x) = \dfrac{5x^2}{\ln(x)}$. This function can be written as $y(x)=\dfrac{u(x)}{v(x)}$, with $u(x)=5x^2$ and $v(x) = \ln(x)$. The derivative of $y(x)$ can be determined as follows (one might consider Derivatives of elementary functions and the example at Sum rule of Product rule):
$$\begin{align} u'(x) &= 5 \cdot 2x = 10x,\\ v'(x) &= \dfrac{1}{x},\\ y'(x) &= \dfrac{u'(x)v(x) + u(x)v'(x)}{\big(v(x)\big)^2} = \dfrac{10x\ln(x) - 5x^2\tfrac{1}{x}}{\ln(x)^2} = \dfrac{10x\ln(x) - 5x}{\ln(x)^2}. \end{align}$$