Questions: Find and simplify the difference quotient (f(x+h)-f(x))/h, h ≠ 0 for the given function. f(x)=1/(7x) A. -(1/(7x(x+h))) B. 0 C. 1/(7x) D. -(1/(x(x+h)))

Solution Steps

To find and simplify the difference quotient for the function \( f(x) = \frac{1}{7x} \), we need to follow these steps:

1. Calculate \( f(x+h) \) by substituting \( x+h \) into the function.
2. Compute the difference \( f(x+h) - f(x) \).
3. Divide the result by \( h \) and simplify the expression.

We start by substituting \( x+h \) into the function \( f(x) = \frac{1}{7x} \): \[ f(x+h) = \frac{1}{7(x+h)} = \frac{1}{7h + 7x} \]

Next, we find the difference \( f(x+h) - f(x) \): \[ f(x+h) - f(x) = \frac{1}{7h + 7x} - \frac{1}{7x} \]

Now, we divide the difference by \( h \): \[ \frac{f(x+h) - f(x)}{h} = \frac{\left(\frac{1}{7h + 7x} - \frac{1}{7x}\right)}{h} \]

To simplify the expression, we combine the fractions in the numerator: \[ \frac{1}{7h + 7x} - \frac{1}{7x} = \frac{7x - (7h + 7x)}{(7h + 7x)(7x)} = \frac{-7h}{(7h + 7x)(7x)} \] Thus, the difference quotient becomes: \[ \frac{-7h}{(7h + 7x)(7x) \cdot h} = \frac{-1}{7x(h + x)} \]

Final Answer