My question is finding the derivitive of
y=((x+λ)^4)/(x^4+λ^4).
I distributed the exponent in the numerator, but I don't know if that's right. Rewritten as y=(x^4+λ^4)/(x^4+λ^4).
I found the derivitive and multiplied accordingly, as
((4x^3+4λ^3)(x^4+λ^4)-(4x^3+4λ^3)(x^4+λ^4))/(x^4+λ^4)^2. I got
y=4x^7+4x^3λ^4+4λ^3x^4+4λ^7-4x^7-4x^3λ^4-4λ^7.
All of the numerator just winds up canceling out and I don't think that's right.
y=(x+λ)^4/(x^4+λ^4)
y' =
4(x+λ)^3(x^4+λ^4) - (x+λ)^4(4x^3)
-----------------------------------------
(x^4+λ^4)^2
=
(4λ)(λ+x)^3(λ^3-x^3)
--------------------------
(x^4+λ^4)^2
(a+b)^4 is NOT a^4+b^4
To find the derivative of the given function, we need to use the quotient rule. Let's break down the steps to get the correct solution:
Step 1: Rewrite the function
You correctly rewrote the function as y = (x^4 + λ^4) / (x^4 + λ^4).
Step 2: Apply the quotient rule
To use the quotient rule, we need to differentiate the numerator and denominator separately, and then apply the rule formula: (f'g - fg') / g^2.
Let's differentiate the numerator (f) and denominator (g):
Numerator (f):
The derivative of x^4 + λ^4 = 4x^3.
Denominator (g):
The derivative of x^4 + λ^4 = 4x^3.
Now we can apply the quotient rule:
y' = [(4x^3)(x^4 + λ^4) - (4x^3)(x^4 + λ^4)] / (x^4 + λ^4)^2
Simplifying the numerator:
y' = 0 / (x^4 + λ^4)^2
Step 3: Simplify the expression
Since the numerator is 0, the derivative of y is 0, which means the function is a constant.
Final Result:
The derivative of y = ((x + λ)^4)/(x^4 + λ^4) is y' = 0.
It's important to note that in this case, the derivative turns out to be zero. This indicates that the function is a constant and does not change with the value of x.