. Find the derivative of the following special functionsb. 𝑦 = ln(4𝑥 2 + 5𝑥 + 3)

To find the derivative of the function y = ln(4x^2 + 5x + 3), we will use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Here, our outer function is ln(u) and our inner function is u = 4x^2 + 5x + 3.

Step 1: Find the derivative of the outer function. The derivative of ln(u) with respect to u is 1/u.

Step 2: Find the derivative of the inner function. The derivative of 4x^2 + 5x + 3 with respect to x is 8x + 5.

Step 3: Apply the chain rule. According to the chain rule, the derivative of y with respect to x is the derivative of the outer function times the derivative of the inner function. So, dy/dx = (1/u) * (8x + 5).

Step 4: Substitute u back into the equation. Substitute u = 4x^2 + 5x + 3 back into the equation to get dy/dx = (8x + 5) / (4x^2 + 5x + 3).

So, the derivative of y = ln(4x^2 + 5x + 3) is dy/dx = (8x + 5) / (4x^2 + 5x + 3).