Question

Find y'. y = [ln(x⁴ + 5)]²

Answer 1

The derivative of y = [ln(x⁴ + 5)]² is found using the chain rule, leading to y' = 8x³ × ln(x⁴ + 5) / (x⁴ + 5).

Step-by-step explanation:

To find the derivative of the function y = [ln(x⁴ + 5)]², we will use the chain rule. The steps to calculate the derivative, y', are as follows:

1. Let u = ln(x⁴ + 5). So, y = u².
2. Derive y with respect to u: dy/du = 2u.
3. Now, derive u with respect to x: du/dx = 1/(x⁴ + 5) × 4x³.
4. Apply the chain rule: dy/dx = dy/du × du/dx.
5. Substitute the expressions for dy/du and du/dx into the chain rule equation: dy/dx = 2 × ln(x⁴ + 5) × 4x³ / (x⁴ + 5).
6. Simplify to get the final result: y' = 8x³ × ln(x⁴ + 5) / (x⁴ + 5).

This is the simplified derivative of the given function y = [ln(x⁴ + 5)]².