07Iquotientrule

toc = The Quotient Rule =


 * Quotient ** means "the result of dividing" so use this when two functions have been divided (or are in a fraction).

math \\ . \qquad \text{If } f(x)=\dfrac{u(x)}{v(x)} \text{, then } \qquad. \\ . \\ . \\ . \qquad \qquad f'(x)=\dfrac{v(x)\cdot{u'(x)}-u(x)\cdot{v'(x)}}{ \big[ v(x) \big]^2} \qquad. math

math \\ . \qquad \text{If } y=\dfrac{u}{v} \text{ where u and v are functions of x then } \qquad .\\. \\ . \\ . \qquad \qquad \dfrac{dy}{dx}=\dfrac{v\cdot{u'}-u\cdot{v}'}{v^2} \qquad. math
 * OR**

** Example 1 **
Find the derivative of each of the following with respect to x: math \text{(a)}\;y=\dfrac{x^{\frac{1}{2}}}{(1+x)} \qquad. \\ . \\ \text{(b)}\;y=\dfrac{log_e(x)}{sin(x)} \qquad. math


 * __Solution:__**

math . \qquad y=\dfrac{x^{\frac{1}{2}}}{(1+x)} \qquad. \\ . \\ . \qquad \quad\;u=x^{\frac{1}{2}}\;\quad\text{and} \quad \;v=1+x \qquad. \\ . \\
 * (a) **

. \qquad \quad\;u'=\dfrac{1}{2}x^{\frac{-1}{2}}\quad\;\text{and}\quad\;v'=1 \qquad. math

math . \qquad \dfrac{dy}{dx}=\dfrac{v\cdot{u'}-u\cdot{v}'}{v^2} \qquad. \\ . \\ . \qquad \qquad=\dfrac{(1+x)\cdot{\frac{1}{2}x^{\frac{-1}{2}}}-x^{\frac{1}{2}}\cdot{1}}{(1+x)^2} \qquad. math

math . \qquad \qquad=\dfrac{{\frac{1}{2}x^{\frac{-1}{2}}(1+x)}-x^{\frac{1}{2}}}{(1+x)^2}\times\dfrac{2x^{\frac{1}{2}}}{2x^{\frac{1}{2}}} \qquad. \\ . \\ . \qquad \qquad=\dfrac{(1+x)-2x}{2x^{\frac{1}{2}}(1+x)^2} \qquad. \\ . \\ .\qquad=\dfrac{1-x}{2x^{\frac{1}{2}}(1+x)^2} \qquad. math

math . \qquad y=\dfrac{\log_e(x)}{\sin(x)} \qquad. \\ . \\ . \qquad \quad\;u=\log_e(x)\ \quad \text{ and }\quad v=\sin(x) \qquad. \\ . \\ . \qquad \quad\;u'=\dfrac{1}{x}\quad\;\text{and}\quad\;v'=cos(x) \qquad. math
 * (b) **

math . \qquad \dfrac{dy}{dx}=\dfrac{v\cdot{u'}-u\cdot{v}'}{v^2} \qquad. \\ . \\ . \qquad \qquad=\dfrac{\sin(x)\cdot{\dfrac{1}{x}}-\log_e(x)\cdot{\cos(x)}}{ \big[ \sin(x) \big]^2} \qquad. \\ . \\ . \qquad \qquad=\dfrac{\sin(x)\cdot{\dfrac{1}{x}} - \log_e(x)\cdot{\cos(x)}}{ \big[ \sin(x) \big]^2 }\times{\dfrac{x}{x}} \qquad. math

math . \qquad \qquad=\dfrac{\sin(x)-x \log_e(x)\cdot{\cos(x)}}{x \sin^2(x)} \qquad. math

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