find y' of the functuion in terms of appropriate variable. simplify as far as possible.
y= ln(1+e^2x)^3
Can you confirm if it is
y=ln[(1+e^(2x))^3]
It never hurts to insert extra parentheses if you do not want unnecessary delay.
y = ln(1+e^2x)^3
= 3ln(ln(1+e^2x)
y' = 3(2e^(2x))/(1 + e^(2x))
= 6e^(2x) /(1 + e^(2x))
To find the derivative (y') of the given function, you need to use the chain rule. The chain rule states that if you have a composite function f(g(x)), then the derivative f'(g(x)) is equal to f'(g(x)) * g'(x).
Let's apply the chain rule to find y' for the given function y = ln((1+e^2x)^3):
Step 1: Identify the outer function
In this case, the outer function is ln(u), where u = (1 + e^2x)^3.
Step 2: Compute the derivative of the outer function
The derivative of ln(u) with respect to u is 1/u.
Step 3: Identify the inner function
The inner function is u = (1 + e^2x)^3.
Step 4: Compute the derivative of the inner function
To find du/dx, we'll apply the chain rule again. Let v = 1 + e^2x, so u = v^3.
Differentiating v with respect to x gives dv/dx = 2e^2x.
Step 5: Apply the chain rule
Using the chain rule, we have:
dy/dx = (dy/du) * (du/dx)
Since dy/du = 1/u and du/dx = 3v^2 * dv/dx (chain rule for v^3), substituting the values, we get:
dy/dx = (1/u) * (3v^2 * dv/dx)
Substituting u and v back, we have:
dy/dx = (1/[(1 + e^2x)^3]) * (3(1 + e^2x)^2 * 2e^2x)
Simplifying further:
dy/dx = 6e^2x / (1 + e^2x)^3
Thus, the derivative of the function y = ln((1+e^2x)^3) is y' = 6e^2x / (1 + e^2x)^3.