Find the derivative of (b/(a+z^2))^2
Assuming that a and b are constants...
f= b(a+z^2)^-2
f'= -2(a+z^2)^-3 * 2z
=-4z/(a+z^2)^3
check that carefully.
How did you get the -2 at the beginning?
f=u^b
f'= b*u^(b-1)
in this case, b=-2
But why is b -2?
Check up the power rule in your notes.
d(x^n)/dx = nx^(n-1)
The power is -2 in the given question.
Remember: "check that carefully."
To find the derivative of the function f(z) = (b/(a+z^2))^2, we can use the chain rule.
First, let's rewrite the function using the power rule and the quotient rule.
f(z) = (b/(a+z^2))^2
= b^2 / (a+z^2)^2
Now, let's apply the chain rule.
Chain Rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
In this case, g(x) = x^2, and h(x) = a + z^2.
So, g'(x) = 2x (by the power rule) and h'(x) = 2z (by the power rule).
Now, we can find the derivative of f(z) by applying the chain rule:
f'(z) = g'(h(z)) * h'(z)
= 2(a+z^2)^2 * 2z
= 4z(a+z^2)(by multiplying terms)
Therefore, the derivative of f(z) = (b/(a+z^2))^2 is given by:
f'(z) = 4z(a+z^2)