Use logarithmic differentiation to find the derivative of the function.
y = (x^4 + 2)^2(x^3 + 4)^4
y = (x^4+4)^2 (x^3+4)^4
ln y = 2ln(x^4+4) + 4ln(x^3+4)
1/y y' = 8x^3/(x^4+4) + 12x^2/(x^3+4)
y' =
[8x^3/(x^4+4) + 12x^2/(x^3+4)]*[(x^4+4)^2 (x^3+4)^4]
Now you can massage that as you will. Eventually you can arrive at
y' = 4x^2 (x^3+4)^3 (x^4+4) (5x^2+8x+12)
To find the derivative of the given function y = (x^4 + 2)^2(x^3 + 4)^4 using logarithmic differentiation, we need to follow these steps:
Step 1: Take the logarithm of both sides of the equation.
ln(y) = ln[(x^4 + 2)^2(x^3 + 4)^4]
Step 2: Apply properties of logarithms to simplify the expression.
ln(y) = 2ln(x^4 + 2) + 4ln(x^3 + 4)
Step 3: Differentiate both sides with respect to x.
d/dx[ln(y)] = d/dx[2ln(x^4 + 2) + 4ln(x^3 + 4)]
Step 4: Use the chain rule to differentiate the individual terms.
(1/y) * dy/dx = 2(1/(x^4 + 2)) * d/dx(x^4 + 2) + 4(1/(x^3 + 4)) * d/dx(x^3 + 4)
Step 5: Simplify the derivatives of the individual terms.
(1/y) * dy/dx = 2(1/(x^4 + 2)) * (4x^3) + 4(1/(x^3 + 4)) * (3x^2)
Step 6: Multiply both sides by y to isolate dy/dx.
dy/dx = y * [2(1/(x^4 + 2)) * (4x^3) + 4(1/(x^3 + 4)) * (3x^2)]
Step 7: Substitute the value of y back into the equation.
dy/dx = [2(1/(x^4 + 2)) * (4x^3) + 4(1/(x^3 + 4)) * (3x^2)] * [(x^4 + 2)^2(x^3 + 4)^4]
Therefore, the derivative of the function y = (x^4 + 2)^2(x^3 + 4)^4 using logarithmic differentiation is dy/dx = [2(1/(x^4 + 2)) * (4x^3) + 4(1/(x^3 + 4)) * (3x^2)] * [(x^4 + 2)^2(x^3 + 4)^4].