Question

Find derivative for 3-5x/5-3x

Answer 1

To find the derivative of the function \( f(x) = \frac{3 - 5x}{5 - 3x} \), we can use the quotient rule. The quotient rule states that if you have a function \( f(x) = \frac{g(x)}{h(x)} \), then the derivative \( f'(x) \) is given by:

\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}
\]

For the function \( f(x) = \frac{3 - 5x}{5 - 3x} \), let:

- \( g(x) = 3 - 5x \)
- \( h(x) = 5 - 3x \)

Now, let's find the derivatives \( g'(x) \) and \( h'(x) \):

1. \( g'(x) = -5 \)
2. \( h'(x) = -3 \)

Now we can apply the quotient rule:

\[
f'(x) = \frac{(-5)(5 - 3x) - (3 - 5x)(-3)}{(5 - 3x)^2}
\]

Now, simplify the numerator:

1. Expand the first term:
\(-5(5 - 3x) = -25 + 15x\)

2. Expand the second term:
\((3 - 5x)(-3) = -9 + 15x\)

Putting it all together:

\[
f'(x) = \frac{-25 + 15x + 9 - 15x}{(5 - 3x)^2}
\]

Combine like terms in the numerator:

\[
f'(x) = \frac{-16}{(5 - 3x)^2}
\]

Thus, the derivative of the function \( f(x) = \frac{3 - 5x}{5 - 3x} \) is:

\[
\boxed{\frac{-16}{(5 - 3x)^2}}
\]