Questions: Find the derivative of the following function. f(x) = (x^3 - 7x^2 + x) / (x - 4) f'(x) =

Solution Steps

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

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

In this case, $g(x) = x^3 - 7x^2 + x$ and $h(x) = x - 4$. We will first find the derivatives $g'(x)$ and $h'(x)$, and then apply the quotient rule.

We start with the function given by

$$f(x) = \frac{x^3 - 7x^2 + x}{x - 4}$$

where $g(x) = x^3 - 7x^2 + x$ and $h(x) = x - 4$.

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

$$g'(x) = 3x^2 - 14x + 1$$ $$h'(x) = 1$$

Using the quotient rule, we find the derivative $f'(x)$:

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

Substituting the values we calculated:

$$f'(x) = \frac{(3x^2 - 14x + 1)(x - 4) - (x^3 - 7x^2 + x)(1)}{(x - 4)^2}$$

After simplifying the expression, we obtain:

$$f'(x) = \frac{2x^3 - 19x^2 + 56x - 4}{(x - 4)^2}$$

Final Answer