Differentiate the following functions
y=In((1-3x^2)/(1+3x^2))^1/2
y=((sin4x)/1+cosx)^5
1) To differentiate the function y = ln((1-3x^2)/(1+3x^2))^(1/2):
Let's simplify the function step by step before differentiating:
1. Apply the power rule for logarithms by bringing the exponent (1/2) down:
y = (1/2)ln((1-3x^2)/(1+3x^2))
2. Apply the chain rule by differentiating the inside function ((1-3x^2)/(1+3x^2)):
y' = (1/2) * 1 / ((1-3x^2)/(1+3x^2)) * ((1+3x^2)/(1-3x^2))'
3. Simplify:
y' = (1/2) * (1+3x^2) / (1-3x^2) * [(1-3x^2)'(1+3x^2) - (1-3x^2)(1+3x^2)'] / (1-3x^2)^2
= (1/2) * (1+3x^2) / (1-3x^2) * [-6x(1+3x^2) - (1-3x^2)6x] / (1-3x^2)^2
= (1/2) * (1+3x^2) / (1-3x^2) * [-6x(1+3x^2) - 6x(1-3x^2)] / (1-3x^2)^2
= (1/2) * (1+3x^2) / (1-3x^2) * (-12x) / (1-3x^2)^2
= -6x(1+3x^2) / [2(1-3x^2)^2]
Therefore, the derivative of y = ln((1-3x^2)/(1+3x^2))^(1/2) is y' = -6x(1+3x^2) / [2(1-3x^2)^2].
2) To differentiate the function y = ((sin4x)/1+cosx)^5:
Let's simplify the function step by step before differentiating:
1. Apply the power rule:
y = (sin4x/(1+cosx))^5
2. Apply the chain rule by differentiating the inside function (sin4x/(1+cosx)):
y' = 5(sin4x / (1+cosx))^4 * (cosx(1+cosx) - sin4x(-sinx)) / (1+cosx)^2
= 5(sin4x)^4 * (cosx(1+cosx) + sin4x(sin4x)) / (1+cosx)^2
= 5(sin^4(4x)) * (cosx(1+cosx) + sin^2(4x)) / (1+cosx)^2
Therefore, the derivative of y = ((sin4x)/1+cosx)^5 is y' = 5(sin^4(4x)) * (cosx(1+cosx) + sin^2(4x)) / (1+cosx)^2.
y = In((1-3x^2)/(1+3x^2))^1/2 = 1/2 ln((1-3x^2)/(1+3x^2))
= 1/2 [ln(1-3x^2) - ln(1+3x^2)]
y' = 1/2 [ -6x/(1-3x^2) - 6x/(1+3x^2)] = 6x/(9x^4 - 1)
Apologies for the confusion in my previous response. You are correct.
The correct differentiation of y = ln((1-3x^2)/(1+3x^2))^1/2 is:
y = (1/2)ln((1-3x^2)/(1+3x^2))
Applying the chain rule:
y' = (1/2) * 1/((1-3x^2)/(1+3x^2)) * ((1-3x^2)/(1+3x^2))'
Simplifying:
y' = (1/2) * (1+3x^2)/(1-3x^2) * [-(1-3x^2)'/(1+3x^2) + (1-3x^2)/(1+3x^2)']
Using the quotient rule:
y' = (1/2) * (1+3x^2)/(1-3x^2) * [(-(-6x))/(1+3x^2) - (1-3x^2)(6x)/(1+3x^2)^2]
Simplifying further:
y' = (1/2) * (1+3x^2)/(1-3x^2) * [(6x/(1+3x^2)) - (6x(1-3x^2))/(1+3x^2)^2]
y' = (1/2) * (1+3x^2)/(1-3x^2) * [(6x - 6x + 18x^3)/(1+3x^2)^2]
y' = (1/2) * (1+3x^2)/(1-3x^2) * (18x^3)/(1+3x^2)^2
simplifying, we have:
y' = 9x^3/(1-9x^4)
To differentiate these functions, we will use the chain rule and basic rules of differentiation. Let's differentiate each function step-by-step:
1. Differentiating y = ln((1 - 3x^2)/(1 + 3x^2))^(1/2):
Step 1: Rewrite the function using exponent properties:
y = (ln((1 - 3x^2)/(1 + 3x^2)))^(1/2)
= ((1 - 3x^2)/(1 + 3x^2))^((1/2) * ln(1 - 3x^2)/(1 + 3x^2))
Step 2: Apply the chain rule by differentiating the outer function and multiplying it by the derivative of the inner function:
dy/dx = (1/2) * ((1 - 3x^2)/(1 + 3x^2))^((1/2) - 1) * (1 - 3x^2)'/(1 + 3x^2) - (1/2) * ((1 - 3x^2)/(1 + 3x^2))^((1/2) * ln(1 - 3x^2)/(1 + 3x^2)) * (ln(1 - 3x^2)/(1 + 3x^2))'
Step 3: Differentiate the terms (1 - 3x^2)' and (ln(1 - 3x^2)/(1 + 3x^2))':
(1 - 3x^2)' = -6x
(ln(1 - 3x^2)/(1 + 3x^2))' = [(1 + 3x^2)(-6x) - (1 - 3x^2)(6x)]/(1 + 3x^2)^2
Step 4: Simplify and combine the terms to find the derivative:
dy/dx = (1/2) * ((1 - 3x^2)/(1 + 3x^2))^(-1/2) * (-6x)/(1 + 3x^2) - (1/2) * ((1 - 3x^2)/(1 + 3x^2))^((1/2) * ln(1 - 3x^2)/(1 + 3x^2)) * [(1 + 3x^2)(-6x) - (1 - 3x^2)(6x)]/(1 + 3x^2)^2
This is the derivative of the function y = ln((1 - 3x^2)/(1 + 3x^2))^(1/2).
2. Differentiating y = ((sin(4x))/(1 + cos(x)))^5:
Step 1: Apply the chain rule by differentiating the outer function and multiplying by the derivative of the inner function:
dy/dx = 5 * (((sin(4x))/(1 + cos(x)))^5-1) * ((sin(4x))/(1 + cos(x)))' * (4x)' - 5 * (((sin(4x))/(1 + cos(x)))^5) * ((sin(4x))/(1 + cos(x)))'
Step 2: Differentiate the terms ((sin(4x))/(1 + cos(x)))' and (4x)':
((sin(4x))/(1 + cos(x)))' = [(cos(4x))(1 + cos(x)) - (sin(4x))(-sin(x))]/(1 + cos(x))^2
(4x)' = 4
Step 3: Simplify and combine the terms to find the derivative:
dy/dx = 5 * (((sin(4x))/(1 + cos(x)))^5-1) * [(cos(4x))(1 + cos(x)) - (sin(4x))(-sin(x))]/(1 + cos(x))^2 * 4 - 5 * (((sin(4x))/(1 + cos(x)))^5) * [(cos(4x))(1 + cos(x)) - (sin(4x))(-sin(x))]/(1 + cos(x))^2
This is the derivative of the function y = ((sin(4x))/(1 + cos(x)))^5.