QuestionJuly 14, 2025

Watch the video and then solve the problem given below. Click here to watch the video. For the function f(x)=(1)/(22x) construct and simplify the difference quotient (f(x+h)-f(x))/(h) The difference quotient for f(x)=(1)/(22x) is square (Simplify your answer. Use integers or fractions for any numbers in the expression.)

Watch the video and then solve the problem given below. Click here to watch the video. For the function f(x)=(1)/(22x) construct and simplify the difference quotient (f(x+h)-f(x))/(h) The difference quotient for f(x)=(1)/(22x) is square (Simplify your answer. Use integers or fractions for any numbers in the expression.)
Watch the video and then solve the problem given below. Click here to watch the video. For the function f(x)=(1)/(22x) construct and simplify the difference quotient (f(x+h)-f(x))/(h) The difference quotient for f(x)=(1)/(22x) is square (Simplify your answer. Use integers or fractions for any numbers in the expression.)

Solution
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Answer

\frac{-1}{22x(x+h)} Explanation 1. Write the difference quotient formula The difference quotient is given by \frac{f(x+h) - f(x)}{h}. 2. Substitute f(x) and f(x+h) For f(x) = \frac{1}{22x}, calculate f(x+h) = \frac{1}{22(x+h)}. Substitute these into the difference quotient: \frac{\frac{1}{22(x+h)} - \frac{1}{22x}}{h}. 3. Simplify the numerator Combine the fractions in the numerator: \frac{1}{22(x+h)} - \frac{1}{22x} = \frac{22x - 22(x+h)}{22x \cdot 22(x+h)} = \frac{-22h}{22^2 x (x+h)}. 4. Simplify the entire expression Divide by h: \frac{-22h}{22^2 x (x+h) \cdot h} = \frac{-22}{22^2 x (x+h)}. 5. Final simplification Simplify the fraction: \frac{-22}{484 x (x+h)} = \frac{-1}{22 x (x+h)}.

Explanation

1. Write the difference quotient formula<br /> The difference quotient is given by $\frac{f(x+h) - f(x)}{h}$.<br /><br />2. Substitute $f(x)$ and $f(x+h)$<br /> For $f(x) = \frac{1}{22x}$, calculate $f(x+h) = \frac{1}{22(x+h)}$. Substitute these into the difference quotient: $\frac{\frac{1}{22(x+h)} - \frac{1}{22x}}{h}$.<br /><br />3. Simplify the numerator<br /> Combine the fractions in the numerator: $\frac{1}{22(x+h)} - \frac{1}{22x} = \frac{22x - 22(x+h)}{22x \cdot 22(x+h)} = \frac{-22h}{22^2 x (x+h)}$.<br /><br />4. Simplify the entire expression<br /> Divide by $h$: $\frac{-22h}{22^2 x (x+h) \cdot h} = \frac{-22}{22^2 x (x+h)}$.<br /><br />5. Final simplification<br /> Simplify the fraction: $\frac{-22}{484 x (x+h)} = \frac{-1}{22 x (x+h)}$.
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