To find and simplify the difference quotient for the given function [tex]\( f(x) = \frac{1}{4x} \)[/tex], follow these detailed steps:
1. Definition of the Difference Quotient:
The difference quotient for any function [tex]\( f(x) \)[/tex] is given by:
[tex]\[
\frac{f(x+h) - f(x)}{h}
\][/tex]
where [tex]\( h \neq 0 \)[/tex].
2. Substitute [tex]\( f(x) \)[/tex] and [tex]\( f(x+h) \)[/tex]:
Here, [tex]\( f(x) = \frac{1}{4x} \)[/tex]. We need to evaluate [tex]\( f(x+h) \)[/tex]:
[tex]\[
f(x + h) = \frac{1}{4(x + h)}
\][/tex]
3. Form the Difference Quotient:
Substitute [tex]\( f(x + h) \)[/tex] and [tex]\( f(x) \)[/tex] into the difference quotient formula:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{\frac{1}{4(x + h)} - \frac{1}{4x}}{h}
\][/tex]
4. Combine the Fractions in the Numerator:
To combine the fractions, find a common denominator:
[tex]\[
\frac{1}{4(x + h)} - \frac{1}{4x} = \frac{x - (x + h)}{4x(x + h)} = \frac{x - x - h}{4x(x + h)} = \frac{-h}{4x(x + h)}
\][/tex]
5. Simplify the Difference Quotient:
Substitute this back into the difference quotient:
[tex]\[
\frac{\frac{-h}{4x(x + h)}}{h} = \frac{-h}{4x(x + h)} \cdot \frac{1}{h} = \frac{-1}{4x(x + h)}
\][/tex]
6. Final Simplified Form:
The simplified form of the difference quotient is:
[tex]\[
\frac{-1}{4x(x + h)}
\][/tex]
So, the correct answer is:
A. [tex]\(\frac{-1}{4x(x + h)}\)[/tex]
