Find the derivative of `(x^(n)-a^(n))/(x-a)` at x=a for some constant a.

Options

A. 1
B. 0
C. `1/2`
D. Does not exist

Explanation

To find the derivative of the function \( f(x) = \frac{x^n - a^n}{x - a} \) at \( x = a \), we can follow these steps:

### Step 1: Identify the Function
We start with the function:
\[ f(x) = \frac{x^n - a^n}{x - a} \]

### Step 2: Apply the Quotient Rule
To differentiate \( f(x) \), we can use the quotient rule, which states:
\[ \left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2} \]
Here, \( u = x^n - a^n \) and \( v = x - a \).

### Step 3: Differentiate \( u \) and \( v \)
Now we calculate the derivatives:
- \( u' = \frac{d}{dx}(x^n - a^n) = n x^{n-1} \) (since \( a^n \) is a constant)
- \( v' = \frac{d}{dx}(x - a) = 1 \)

### Step 4: Substitute into the Quotient Rule
Now we substitute \( u \), \( u' \), \( v \), and \( v' \) into the quotient rule:
\[ f'(x) = \frac{(x - a)(n x^{n-1}) - (x^n - a^n)(1)}{(x - a)^2} \]

### Step 5: Simplify the Expression
This simplifies to:
\[ f'(x) = \frac{n x^{n-1} (x - a) - (x^n - a^n)}{(x - a)^2} \]

### Step 6: Evaluate at \( x = a \)
Now we need to find \( f'(a) \):
\[ f'(a) = \frac{n a^{n-1} (a - a) - (a^n - a^n)}{(a - a)^2} \]
This results in:
\[ f'(a) = \frac{0 - 0}{0} = \frac{0}{0} \]
This is an indeterminate form.

### Step 7: Apply L'Hôpital's Rule
Since we have an indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule, which states that we can take the derivative of the numerator and the derivative of the denominator separately:
1. Differentiate the numerator \( x^n - a^n \) to get \( n x^{n-1} \).
2. Differentiate the denominator \( x - a \) to get \( 1 \).

Now we can evaluate the limit:
\[ \lim_{x \to a} f'(x) = \lim_{x \to a} \frac{n x^{n-1}}{1} = n a^{n-1} \]

### Conclusion
Thus, the derivative of \( f(x) \) at \( x = a \) is:
\[ f'(a) = n a^{n-1} \]