Question

Use logarithmic differentiation to find the derivative of the function. y = (x³ + 2)²(x⁴ + 4)⁴

Answer 1

To find the derivative of the function y = (x³ + 2)²(x⁴ + 4)⁴ using logarithmic differentiation, follow these steps: take the natural logarithm of both sides, apply the properties of logarithms, differentiate each term using the chain rule, and simplify the expression to find the derivative.

Step-by-step explanation:

To find the derivative of the function y = (x³ + 2)²(x⁴ + 4)⁴ using logarithmic differentiation, we can take the natural logarithm of both sides of the equation. This allows us to simplify the expression and differentiate it with respect to x. Here are the steps:

1. Take the natural logarithm (ln) of both sides: ln(y) = ln((x³ + 2)²(x⁴ + 4)⁴)
2. Apply the properties of logarithms to expand the equation: ln(y) = ln((x³ + 2)²) + ln((x⁴ + 4)⁴)
3. Use the chain rule to differentiate each term: 1/y * y' = 2(x³ + 2)(3x²) + 4(x⁴ + 4)³(4x³)
4. Simplify the expression and solve for y': y' = y * [2(x³ + 2)(3x²) + 4(x⁴ + 4)³(4x³)]

Therefore, the derivative of the function y = (x³ + 2)²(x⁴ + 4)⁴ is y' = y * [2(x³ + 2)(3x²) + 4(x⁴ + 4)³(4x³)].