Find the𝑦 = ln(3𝑥 2 + 2𝑥) 2 derivative of the following special functions

To find the derivative of the function y = ln(3x^2 + 2x)^2, we will use the chain rule and the power rule.

Step 1: Identify the outer function and the inner function. In this case, the outer function is y = ln(u)^2 and the inner function is u = 3x^2 + 2x.

Step 2: Differentiate the outer function with respect to u. The derivative of y = ln(u)^2 with respect to u is (2/u) * (1/ln(10)).

Step 3: Differentiate the inner function with respect to x. The derivative of u = 3x^2 + 2x with respect to x is 6x + 2.

Step 4: Multiply the derivatives from steps 2 and 3 together. The derivative of y = ln(3x^2 + 2x)^2 with respect to x is (2/u) * (1/ln(10)) * (6x + 2).

Step 5: Substitute u = 3x^2 + 2x back into the derivative. The derivative of y = ln(3x^2 + 2x)^2 with respect to x is (2/(3x^2 + 2x)) * (1/ln(10)) * (6x + 2).

So, the derivative of y = ln(3x^2 + 2x)^2 is (2/(3x^2 + 2x)) * (1/ln(10)) * (6x + 2).