Question

Use the product rule or the quotient rule to find the derivative.

(a) y=(2 x+1) / (x³}+4 x² - 7 x+11
(b) y={3 x+4} / {2 x⁴ -7} y={z}= quotient

Answer 1

The derivative of y=(2x+1)/(x³+4x²-7x+11) is found using the quotient rule, resulting in y' = [(x³ + 4x² - 7x + 11)*(2) - (2x + 1)*(3x² + 8x - 7)]/(x³ + 4x² - 7x + 11)² after simplification.

Step-by-step explanation:

To find the derivative of the given function y=(2x+1)/(x³+4x²-7x+11), we need to apply the quotient rule. The quotient rule states that if you have a function that is the ratio of two functions, u(x)/v(x), its derivative y' is given by (v(x)*u'(x) - u(x)*v'(x))/(v(x))^2.

Let's denote the numerator function as u(x) = 2x + 1 and the denominator as v(x) = x³ + 4x² - 7x + 11. We compute the derivatives u'(x) and v'(x) using the power rule. Now apply the quotient rule:

y' = (v(x)*u'(x) - u(x)*v'(x))/(v(x))² = [(x³ + 4x² - 7x + 11)*(2) - (2x + 1)*(3x² + 8x - 7)]/(x³ + 4x² - 7x + 11)²

Simplify the expression to get the final derivative.

y' = [(x³ + 4x² - 7x + 11)*(2) - (2x + 1)*(3x² + 8x - 7)]/(x³ + 4x² - 7x + 11)²