If `Q=(x^n)/(y^m) and Deltax, Deltay` are absolute errors in the measurement of x and y then absolute error `DeltaQ ` in Q is

Options

A. `DeltaQ =pm (n(Deltax)/x+m(Deltay)/y)`
B. `DeltaQ=pm(n(Deltax)/x+m(Deltay)/y)Q`
C. `DeltaQ=pm(n(Deltax)/x-m(Deltay)/y)Q`
D. `DeltaQ=pm(n(Deltax)/y-m(Deltay)/x)Q`

Answer

To find the absolute error \(\Delta Q\) in the quantity \(Q = \frac{x^n}{y^m}\), we will use the concept of propagation of errors. Here are the steps to derive the expression for \(\Delta Q\):

### Step 1: Write the expression for \(Q\)
We start with the expression for \(Q\):
\[ Q = \frac{x^n}{y^m} \]

### Step 2: Rewrite \(Q\) using properties of exponents
We can rewrite \(Q\) as:
\[ Q = x^n \cdot y^{-m} \]

### Step 3: Apply the product rule for derivatives
To find the absolute error \(\Delta Q\), we need to differentiate \(Q\) with respect to \(x\) and \(y\). The total differential \(dQ\) is given by:
\[ dQ = \frac{\partial Q}{\partial x} dx + \frac{\partial Q}{\partial y} dy \]

### Step 4: Calculate the partial derivatives
1. The partial derivative with respect to \(x\):
\[ \frac{\partial Q}{\partial x} = n \cdot x^{n-1} \cdot y^{-m} \]
2. The partial derivative with respect to \(y\):
\[ \frac{\partial Q}{\partial y} = -m \cdot x^n \cdot y^{-m-1} \]

### Step 5: Substitute the absolute errors
Now, substituting \(dx = \Delta x\) and \(dy = \Delta y\) into the total differential:
\[ dQ = n \cdot x^{n-1} \cdot y^{-m} \Delta x - m \cdot x^n \cdot y^{-m-1} \Delta y \]

### Step 6: Express \(\Delta Q\) in terms of \(Q\)
Now, we can express \(\Delta Q\) as:
\[ \Delta Q = dQ = Q \left( \frac{n \Delta x}{x} - \frac{m \Delta y}{y} \right) \]
where \(Q = \frac{x^n}{y^m}\).

### Step 7: Write the final expression for absolute error
Thus, the absolute error in \(Q\) can be written as:
\[ \Delta Q = Q \left( \frac{n \Delta x}{x} - \frac{m \Delta y}{y} \right) \]

### Step 8: Include the sign for absolute error
Since we are dealing with absolute errors, we include the \(\pm\) sign:
\[ \Delta Q = \pm Q \left( \frac{n \Delta x}{x} - \frac{m \Delta y}{y} \right) \]

### Final Answer
The absolute error \(\Delta Q\) in \(Q\) is:
\[ \Delta Q = \pm Q \left( \frac{n \Delta x}{x} - \frac{m \Delta y}{y} \right) \]