Find the derivative of the function.
G(y) =
(y − 1)3/
(y2 + 2y)7
To find the derivative of the function G(y), we can use the quotient rule. The quotient rule states that if we have a function in the form f(x) = u(x)/v(x), then its derivative is given by:
f'(x) = (v(x)*u'(x) - u(x)*v'(x)) / (v(x))^2
In this case, u(x) = (y − 1)^3 and v(x) = (y^2 + 2y)^7. Let's find the derivative step by step:
Step 1: Find u'(x)
To find the derivative of u(x), we apply the power rule for differentiation:
u'(x) = 3(y − 1)^(3-1) * (1)
Simplifying, we have u'(x) = 3(y − 1)^2.
Step 2: Find v'(x)
To find the derivative of v(x), we again apply the power rule for differentiation:
v'(x) = 7(y^2 + 2y)^(7-1) * (2y + 2)
Simplifying, we have v'(x) = 14(y^2 + 2y)^6 * (y + 1).
Step 3: Calculate the derivative of G(y)
Using the quotient rule, we substitute u(x), v(x), u'(x), and v'(x) into the formula:
G'(y) = [(v(x)*u'(x) - u(x)*v'(x))] / (v(x))^2
G'(y) = [(y^2 + 2y)^7 * 3(y − 1)^2 - (y − 1)^3 * 14(y^2 + 2y)^6 * (y + 1)] / [(y^2 + 2y)^7]^2
Simplifying this expression further may be difficult due to the complexity of the terms involved. However, you can use this formula to calculate the derivative of the function G(y) using a symbolic algebra system or a graphing calculator with a derivative function.
Assuming you mean
G(y) = (y-1)^3 / (y^2+2)^7
then just use the quotient rule to get
-(y-1)^2 (11y^2-14y-6) / (y^2+2)^8