Let `y = 1 + (a_(1))/(x - a_(1)) + (a_(2) x)/((x - a_(1))(x - a_(2))) + (a_(3) x^(2))/((x - a_(1))(x - a_(2))(x - a_(3))) + … (a_(n) x^(n - 1))/((x - a_(1))(x - a_(2))(x - a_(3))..(x - a_(n)))` Find `(dy)/(dx)`

Answer

To find the derivative \( \frac{dy}{dx} \) of the given function

\[ y = 1 + \frac{a_1}{x - a_1} + \frac{a_2 x}{(x - a_1)(x - a_2)} + \frac{a_3 x^2}{(x - a_1)(x - a_2)(x - a_3)} + \ldots + \frac{a_n x^{n-1}}{(x - a_1)(x - a_2)(x - a_3) \ldots (x - a_n)}, \]

we can follow these steps:

### Step 1: Rewrite the Function
We can express \( y \) in a more compact form. Notice that the terms can be rewritten as:

\[ y = \sum_{k=0}^{n} \frac{a_k x^{k-1}}{\prod_{j=1}^{k}(x - a_j)} \]

for \( k = 1 \) to \( n \). The first term is simply 1.

### Step 2: Identify the Structure
We can factor out common terms from the series. The structure of \( y \) suggests that it can be expressed as:

\[ y = \frac{x^n}{\prod_{j=1}^{n}(x - a_j)}. \]

### Step 3: Take the Natural Logarithm
To differentiate, we can take the natural logarithm of both sides:

\[ \ln y = \ln \left( \frac{x^n}{\prod_{j=1}^{n}(x - a_j)} \right). \]

Using properties of logarithms, we can rewrite this as:

\[ \ln y = n \ln x - \sum_{j=1}^{n} \ln(x - a_j). \]

### Step 4: Differentiate Both Sides
Now, we differentiate both sides with respect to \( x \):

\[ \frac{1}{y} \frac{dy}{dx} = \frac{n}{x} - \sum_{j=1}^{n} \frac{1}{x - a_j} \cdot \frac{d}{dx}(x - a_j). \]

Since \( \frac{d}{dx}(x - a_j) = 1 \), we have:

\[ \frac{1}{y} \frac{dy}{dx} = \frac{n}{x} - \sum_{j=1}^{n} \frac{1}{x - a_j}. \]

### Step 5: Solve for \( \frac{dy}{dx} \)
Multiplying through by \( y \):

\[ \frac{dy}{dx} = y \left( \frac{n}{x} - \sum_{j=1}^{n} \frac{1}{x - a_j} \right). \]

### Final Expression
Substituting back for \( y \):

\[ \frac{dy}{dx} = \frac{x^n}{\prod_{j=1}^{n}(x - a_j)} \left( \frac{n}{x} - \sum_{j=1}^{n} \frac{1}{x - a_j} \right). \]

### Summary
Thus, the derivative \( \frac{dy}{dx} \) is given by:

\[ \frac{dy}{dx} = \frac{x^n}{(x - a_1)(x - a_2) \ldots (x - a_n)} \left( \frac{n}{x} - \left( \frac{1}{x - a_1} + \frac{1}{x - a_2} + \ldots + \frac{1}{x - a_n} \right) \right). \]