Question

Differentiate y=√64−x² .⋅8arcsin(x³/8).

Answer 1

To differentiate the given function, apply the chain rule and product rule. The derivative of y is -8x * arcsin(x³/8) + 3x²√(64-x²)/8.

Step-by-step explanation:

To differentiate the given function, we will need to apply the chain rule and product rule. Let's break down the steps:

1. First, differentiate the square root function. The derivative of √(64-x²) is -x/√(64-x²).
2. Next, differentiate the arcsin function. The derivative of arcsin(x³/8) is (3x²)/[8√(64-x²)].

3. -x/√(64-x²) * 8arcsin(x³/8) + √(64-x²) * (3x²)/[8√(64-x²)].

Simplifying the expression, we get:

dy/dx = -8x * arcsin(x³/8) + 3x²√(64-x²)/8.