Questions: Question 17 Differentiate. Find f'(t) for f(x) = x/(8x-8) (Use the Quotient Rule.) -8x/(8x-8)^2 -8/(8x-8)^2 -8/(8x-8) (16x-8)/(8x-8)^2

Transcript text: Question 17 Differentiate. Find $f^{\prime}(t)$ for $f(x)=\frac{x}{8 x-8}$ (Use the Quotient Rule.) $-\frac{8 x}{(8 x-8)^{2}}$ $-\frac{8}{(8 x-8)^{2}}$ $-\frac{8}{8 x-8}$ $\frac{16 x-8}{(8 x-8)^{2}}$

Solution

Solution Steps

Solution Approach

To differentiate the function $ f(x) = \frac{x}{8x - 8} $ using the Quotient Rule, we need to apply the formula for the derivative of a quotient of two functions. The Quotient Rule states that if $ f(x) = \frac{g(x)}{h(x)} $, then $ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} $. Here, $ g(x) = x $ and $ h(x) = 8x - 8 $. We will find the derivatives $ g'(x) $ and $ h'(x) $, and then apply the Quotient Rule formula.

Step 1: Define the Functions

We start with the function $ f(x) = \frac{x}{8x - 8} $. Here, we identify the numerator and denominator:

$ g(x) = x $
$ h(x) = 8x - 8 $

Step 2: Compute the Derivatives

Next, we compute the derivatives of $ g(x) $ and $ h(x) $:

$ g'(x) = 1 $
$ h'(x) = 8 $

Step 3: Apply the Quotient Rule

Using the Quotient Rule, which states that if $ f(x) = \frac{g(x)}{h(x)} $, then $ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} $, we substitute our values: \[ f'(x) = \frac{1 \cdot (8x - 8) - x \cdot 8}{(8x - 8)^2} \]

Step 4: Simplify the Expression

Now, we simplify the expression: \[ f'(x) = \frac{8x - 8 - 8x}{(8x - 8)^2} = \frac{-8}{(8x - 8)^2} \] This can be further simplified to: \[ f'(x) = -\frac{1}{8(x - 1)^2} \]