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Solution by Steps

step 1

To differentiate the function

\frac{Y^2}{x}

with respect to

x

, we apply the quotient rule

step 2

The quotient rule states that

\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

, where

f = Y^2

and

g = x

. Since

Y

is treated as a constant with respect to

x

,

f' = 0

step 3

Applying the quotient rule, we get

\frac{d}{dx}\left(\frac{Y^2}{x}\right) = \frac{0 \cdot x - Y^2 \cdot 1}{x^2}

step 4

Simplifying the expression, we find

\frac{-Y^2}{x^2}

Answer

\frac{d}{dx}\left(\frac{Y^2}{x}\right) = -\frac{Y^2}{x^2}

Key Concept

Differentiation using the quotient rule

Explanation

The quotient rule is used to differentiate a function that is the quotient of two other functions, particularly when the numerator is a function of a constant with respect to the variable of differentiation.

use an augmented matrix to solve the system x+2y-3z=12 5x-3y+z=-11 2x+y+4z=-5

Solution by Steps

step 1

Convert the system of equations into an augmented matrix:

$$ \begin{bmatrix} 1 & 2 & -3 & | & 12 \\ 5 & -3 & 1 & | & -11 \\ 2 & 1 & 4 & | & -5 \end{bmatrix} $$

step 2

Perform row reduction on the augmented matrix to reach row-echelon form:

$$ \begin{bmatrix} 1 & 0 & 0 & | & 0 \\ 0 & 1 & 0 & | & 3 \\ 0 & 0 & 1 & | & -2 \end{bmatrix} $$

step 3

Interpret the row-reduced matrix as a system of equations to find the solution:

$$ \begin{cases} x = 0 \\ y = 3 \\ z = -2 \end{cases} $$

Answer

The solution to the system of equations is

x = 0

,

y = 3

, and

z = -2

.

Key Concept

Row Reduction of Augmented Matrices

Explanation

The row reduction process transforms the augmented matrix of a system of linear equations into a form where the solution can be easily read off as the entries in the last column.

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