Find the derivative.
y=e^8x/[e^(8x)+9]
y' =
The answer I have is 8e^8x/[(e^8x+9)^2] but it marked it incorrect..
okay here
set u=e^(8x)
v=(e^(8x)+9)
du/dx=8e^(8x)
dv/dx=8e^8x
dy/dx=[(e^8x+9)8e^8x-e^8x(8e^8x)]/(e^8x+9)^2
dy/dx=(8e^64x+72e^8x-8e^64x]/(e^8x+9)^2
=72e^8x/(e^8x+9)^2
check if i mad any erroe
I guess you did.
y' =
(8e^(8x))(e^(8x)+9)-(e^(8x))(8e^(8x))
-------------------------------
(e^(8x)+9)^2
= 72e^(8x)/(e^(8x)+9)^2
You can use wolframalpha.com to confirm your results:
http://www.wolframalpha.com/input/?i=derivative+e%5E(8x)%2F%5Be%5E(8x)%2B9%5D
To find the derivative of the given function, we can use the quotient rule. The quotient rule states that if we have a function in the form of u/v, where u and v are both functions of x, then the derivative of the function is given by:
(f/g)' = (g * f' - f * g') / (g^2)
Let's apply this to the given function:
Given function: y = e^(8x) / (e^(8x) + 9)
To find the derivative, we need to find y':
Step 1: Find the derivative of the numerator (u'):
For u = e^(8x), the derivative is u' = (e^(8x))' = 8e^(8x)
Step 2: Find the derivative of the denominator (v'):
For v = e^(8x) + 9, the derivative is v' = (e^(8x) + 9)'
To find the derivative of e^(8x), we can use the chain rule. The chain rule states that if we have a function of the form f(g(x)), then the derivative is given by f'(g(x)) * g'(x). In this case, f(x) = e^x and g(x) = 8x.
So, applying the chain rule, the derivative of e^(8x) is (e^(8x))' = (e^x) * (8) = 8e^(8x)
Since the derivative of a constant (9 in this case) is zero, the derivative of v = e^(8x) + 9 is v' = 8e^(8x)
Step 3: Apply the quotient rule formula:
(f/g)' = (g * f' - f * g') / (g^2)
Substituting the values we found, we have:
y' = [(e^(8x) + 9) * 8e^(8x) - e^(8x) * 8e^(8x)] / (e^(8x) + 9)^2
Simplifying this expression further:
y' = [8e^(16x) + 72e^(8x) - 8e^(16x)] / (e^(8x) + 9)^2
y' = 72e^(8x) / (e^(8x) + 9)^2
So, the correct answer for y' is 72e^(8x) / (e^(8x) + 9)^2
Double-check the calculation steps to ensure accuracy.