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