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To find the derivative \( f'(x) \) of the function 

\[
f(x) = \frac{bx + c}{kx + a},
\]

we can apply the **quotient rule** for derivatives. Recall that if \( f(x) = \frac{u(x)}{v(x)} \), then 

\[
f'(x) = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}.
\]

Here, we have:

- \( u(x) = bx + c \) with \( u'(x) = b \),
- \( v(x) = kx + a \) with \( v'(x) = k \).

Using the quotient rule, we get:

\[
f'(x) = \frac{(bx + c)'(kx + a) - (bx + c)(kx + a)'}{(kx + a)^2}.
\]

Substitute \( u'(x) = b \) and \( v'(x) = k \):

\[
f'(x) = \frac{b(kx + a) - (bx + c)k}{(kx + a)^2}.
\]

Expanding and simplifying the numerator:

\[
= \frac{bkx + ba - bkx - ck}{(kx + a)^2}.
\]

Now, notice that \( bkx \) terms cancel out:

\[
= \frac{ba - ck}{(kx + a)^2}.
\]

Therefore, the derivative is:

\[
f'(x) = \frac{ba - ck}{(kx + a)^2}.
\]

Would you like a more detailed breakdown, or do you have any questions?

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Here are some related questions for further understanding:

1. How would the answer change if \( b = k \)?
2. What happens to the derivative if \( c = 0 \)?
3. Can you apply the quotient rule to functions where both numerator and denominator are non-linear?
4. How does the derivative behave as \( x \to \infty \)?
5. What if \( a = 0 \)? How does it affect \( f(x) \) and \( f'(x) \)?

**Tip:** When applying the quotient rule, carefully simplify to see if terms cancel out; this often reduces the complexity of your answer.

Find the derivative of f(x) = (bx + c) / (kx + a). Assume that a, b, c, and k are constants.

Learn how to find the derivative of the function f(x) = (bx + c) / (kx + a) using the quotient rule. This problem covers key calculus concepts, including the application of the quotient rule to differentiate a rational function. Suitable for advanced high school students in Grades 11-12.

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