Find Derivative of x/y [x divided by y]

The derivative of x/y with respect to x is equal to (-x dy/dx + y)/y². Here, we will learn how to differentiate x/y with respect to x.

Derivative of x/y with respect to x

Question: Differentiate x/y with respect to x.

Solution:

Note that ${\displaystyle \frac{x}{y}}$ is a product of two functions $x$ and ${\displaystyle \frac{1}{y}}$. So to find its derivative with respect to $x$, we will use the product rule of derivatives.

We have

${\displaystyle \frac{d}{dx}}({\displaystyle \frac{x}{y}})$ $={\displaystyle \frac{d}{dx}}(x\cdot {\displaystyle \frac{1}{y}})$

= $x\cdot {\displaystyle \frac{d}{dx}}({\displaystyle \frac{1}{y}})+{\displaystyle \frac{1}{y}}\cdot {\displaystyle \frac{d}{dx}}(x)$

= $x\cdot {\displaystyle \frac{d}{dx}}(y^{-1})+{\displaystyle \frac{1}{y}}\cdot 1$

= $x\cdot (-1\cdot y^{-1-1}){\displaystyle \frac{dy}{dx}}+{\displaystyle \frac{1}{y}}$, by the power rule and the chain rule of derivatives.

= $-x\cdot y^{-2}{\displaystyle \frac{dy}{dx}}+{\displaystyle \frac{1}{y}}$

= $-{\displaystyle \frac{x}{y^2}}{\displaystyle \frac{dy}{dx}}+{\displaystyle \frac{1}{y}}$

= ${\displaystyle \frac{-x{\displaystyle \frac{dy}{dx}}+y}{y^2}}$

So the differentiation of x/y with respect to x is equal to (-x dy/dx + y)/y².

Derivative of x/y with respect to y

Question: Differentiate x/y with respect to y.

Solution:

In a similar way as above, we have that

${\displaystyle \frac{d}{dy}}({\displaystyle \frac{x}{y}})$ $={\displaystyle \frac{d}{dy}}(x\cdot {\displaystyle \frac{1}{y}})$

= $x\cdot {\displaystyle \frac{d}{dy}}({\displaystyle \frac{1}{y}})+{\displaystyle \frac{1}{y}}\cdot {\displaystyle \frac{d}{dy}}(x)$

= $x\cdot {\displaystyle \frac{d}{dy}}(y^{-1})+{\displaystyle \frac{1}{y}}{\displaystyle \frac{dx}{dy}}$

= $x\cdot (-1\cdot y^{-1-1})+{\displaystyle \frac{1}{y}}{\displaystyle \frac{dx}{dy}}$, by the power rule of derivatives.

= ${\displaystyle \frac{-x}{y^2}}+{\displaystyle \frac{1}{y}}{\displaystyle \frac{dx}{dy}}$

= ${\displaystyle \frac{-x+y\frac{dx}{dy}}{y^2}}$

Hence the differentiation of x/y with respect to y is equal to (-x+ y dx/dy)/y².

Also Read: Derivative of Square Root of 2x

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