k(z)= 4z^4 - z(z+10)/((1+z)^2)
derivative?
16z^3 + (3z^3 + 26z^2 + 42z + 10)/((1+z)^4)
this is wrong though ^
To find the derivative of the given function, we can use the power rule and the quotient rule of differentiation.
The power rule states that if we have a function of the form f(x) = ax^n, then its derivative is given by f'(x) = nax^(n-1).
The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then its derivative is given by f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^2.
Let's find the derivative step by step:
Given function: k(z) = 4z^4 - z(z+10)/((1+z)^2)
Step 1: Expand the expression and simplify as much as possible:
k(z) = 4z^4 - z^2 - 10z/((1+z)^2)
= 4z^4 - z^2 - 10z/(1+2z+z^2)
Step 2: Differentiate the function using the power rule and the quotient rule:
k'(z) = (d/dz)(4z^4) - (d/dz)(z^2) - (d/dz)(10z/(1+2z+z^2))
Applying the power rule:
k'(z) = 16z^3 - 2z
For the last term, we need to use the quotient rule:
k'(z) = 16z^3 - 2z - (d/dz)(10z/(1+2z+z^2))
= 16z^3 - 2z - (10(1+2z+z^2) - 10z(2+2z))/(1+2z+z^2)^2
= 16z^3 - 2z - (10 + 20z + 10z^2 - 20z - 20z^2)/(1+2z+z^2)^2
= 16z^3 - 2z - (10 - 10z^2 - 10z)/(1+2z+z^2)^2
= 16z^3 - 2z - (10 - 10z(z+1))/(1+2z+z^2)^2
= 16z^3 - 2z - (10 - 10z^2 - 10z)/(1+2z+z^2)^2
= 16z^3 - 2z - (10z^2 + 10z - 10)/(1+2z+z^2)^2
Therefore, the derivative of k(z) is:
k'(z) = 16z^3 - 2z - (10z^2 + 10z - 10)/(1+2z+z^2)^2