f'(x) = (8x - 11) / (x^2 - 3x + 2)^2

<p>To find the derivative of f(x), we will use the quotient rule:</p><p>f(x) = (x^3 - 2x^2 + 4x - 7) / (x^2 - 3x + 2)</p><p>f'(x) = [ (x^2 - 3x + 2)(3x^2 - 4x + 4) - (x^3 - 2x^2 + 4x - 7)(2x - 3) ] / (x^2 - 3x + 2)^2</p><p>Simplifying this expression gives:</p><p>f'(x) = [ 3x^4 - 14x^3 + 21x^2 - 2x - 22 ] / (x^2 - 3x + 2)^2</p><p>Therefore, the derivative of f(x) is:</p><p>f'(x) = [ 3x^4 - 14x^3 + 21x^2 - 2x - 22 ] / (x^2 - 3x + 2)^2</p>

<p>To find the derivative of the function -f(x) = (x^3 - 2x^2 + 4x - 7) / (x^2 - 3x + 2), we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x) / h(x), where g(x) and h(x) are differentiable functions, then the derivative of f(x) is given by:</p><p><br></p><p>f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]^2</p><p><br></p><p>In our case, g(x) = x^3 - 2x^2 + 4x - 7 and h(x) = x^2 - 3x + 2. Let's differentiate both g(x) and h(x) to apply the quotient rule.</p><p><br></p><p>First, let's find g'(x), the derivative of g(x):</p><p>g'(x) = d/dx (x^3 - 2x^2 + 4x - 7)</p><p>= 3x^2 - 4x + 4</p><p><br></p><p>Next, let's find h'(x), the derivative of h(x):</p><p>h'(x) = d/dx (x^2 - 3x + 2)</p><p>= 2x - 3</p><p><br></p><p>Now, we can apply the quotient rule to find the derivative of -f(x):</p><p>-f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]^2</p><p>= [(3x^2 - 4x + 4) * (x^2 - 3x + 2) - (x^3 - 2x^2 + 4x - 7) * (2x - 3)] / (x^2 - 3x + 2)^2</p><p><br></p><p>Simplifying this expression would give the derivative of -f(x) with respect to x.</p><p><br></p><p>-f'(x) = [(3x^2 - 4x + 4) * (x^2 - 3x + 2) - (x^3 - 2x^2 + 4x - 7) * (2x - 3)] / (x^2 - 3x + 2)^2</p><p><br></p><p>Expanding the terms:</p><p>-f'(x) = [3x^4 - 4x^3 + 4x^2 - 9x^3 + 12x^2 - 12x + 6x^3 - 8x^2 + 8x - 14 - (2x^4 - 4x^3 + 8x^2 - 12x^2 + 6x^3 - 12x + 8x^2 - 16x + 21)] / (x^2 - 3x + 2)^2</p><p><br></p><p>Simplifying the expression:</p><p>-f'(x) = [x^4 - 3x^3 + 8x - 14 - 2x^4 + 3x^3 - 16x + 21] / (x^2 - 3x + 2)^2</p><p>= [-x^4 + 8x - 7] / (x^2 - 3x + 2)^2</p><p><br></p><p>Therefore, the derivative of -f(x) = (x^3 - 2x^2 + 4x - 7) / (x^2 - 3x + 2) is given by -f'(x) = (-x^4 + 8x - 7) / (x^2 - 3x + 2)^2.</p>