Final answer:
The derivative h'(x) for the function h(x) = g(x) / f(x), with given functions f(x) and g(x), is found using the quotient rule. The derivatives g'(x) and f'(x) are calculated, substituted into the quotient rule formula, and simplified to obtain h'(x). Finally, h'(1) is calculated by substituting x = 1 into h'(x).
Step-by-step explanation:
To find the derivative h'(x) of the function h(x) = g(x) / f(x), where f(x) = 2x² + x - 3 and g(x) = -3x² - 4x + 2, we can use the quotient rule for differentiation. The quotient rule states that the derivative of a function in the form u(x)/v(x) is (v(x)u'(x) - u(x)v'(x)) / v(x)². In this case, u(x) = g(x) and v(x) = f(x).
First, we find the derivatives of g(x) and f(x):
g'(x) = d(-3x² - 4x + 2)/dx = -6x - 4
f'(x) = d(2x² + x - 3)/dx = 4x + 1
Next, applying the quotient rule:
h'(x) = (f(x)g'(x) - g(x)f'(x)) / f(x)² = ((2x² + x - 3)(-6x - 4) - (-3x² - 4x + 2)(4x + 1)) / (2x² + x - 3)²
To find h'(1), we substitute x = 1 into the derivative:
h'(1) = ((2(1)² + 1 - 3)(-6(1) - 4) - (-3(1)² - 4(1) + 2)(4(1) + 1)) / (2(1)² + 1 - 3)²
Therefore, h'(1) is the value obtained after calculating the above expression.