Question

Find the derivative dy/dx for y= x² + 4x / x³ - 5

Answer 1

The derivative of y = (x² + 4x) / (x³ - 5) is (-10x² + 8x - 12x⁴ + 24x²) / (x³ - 5)²

Step-by-step explanation:

To find the derivative of y = (x² + 4x) / (x³ - 5), we can use the quotient rule.

The quotient rule states that if y = u / v, then the derivative dy/dx is given by (v * du/dx - u * dv/dx) / (v²).

Let's apply the quotient rule to the given function:

1. u = x² + 4x
2. v = x³ - 5

Now, let's find du/dx and dv/dx:

1. du/dx = 2x + 4
2. dv/dx = 3x²

Next, substitute these values into the quotient rule:

dy/dx = ((x³ - 5) * (2x + 4) - (x² + 4x) * 3x²) / (x³ - 5)²

Simplify the expression to get the final derivative:

dy/dx = (2x⁴ - 10x² + 8x - 12x⁴ + 24x²) / (x³ - 5)²

dy/dx = (-10x² + 8x - 12x⁴ + 24x²) / (x³ - 5)²