Use logarithmic differentiation to find the derivative of the function.
y = (x^4 + 2)^2(x^3 + 4)^4
To find the derivative of the function using logarithmic differentiation, follow these steps:
1. Take the natural logarithm (ln) of both sides of the equation:
ln(y) = ln((x^4 + 2)^2(x^3 + 4)^4)
2. Use the logarithmic properties to simplify the expression. Apply the power rule of logarithms and distribute the exponents:
ln(y) = 2ln(x^4 + 2) + 4ln(x^3 + 4)
3. Differentiate both sides of the equation with respect to x. Remember that ln(y) represents the natural logarithm of y with respect to x:
(1/y) * y' = 2(1/(x^4 + 2)) * (4x^3) + 4(1/(x^3 + 4)) * (3x^2)
4. Simplify the expression:
y' = y * [2(4x^3) / (x^4 + 2) + 4(3x^2) / (x^3 + 4)]
5. Substitute back the original expression for y:
y' = (x^4 + 2)^2(x^3 + 4)^4 * [2(4x^3) / (x^4 + 2) + 4(3x^2) / (x^3 + 4)]
Therefore, the derivative of the function y = (x^4 + 2)^2(x^3 + 4)^4 is given by y' = (x^4 + 2)^2(x^3 + 4)^4 * [2(4x^3) / (x^4 + 2) + 4(3x^2) / (x^3 + 4)].