If y=ln((3x+2)^6(x+10)^3), then dy/dx=

To find the derivative of the function y = ln((3x + 2)^6(x + 10)^3), we can use the chain rule and the power rule for logarithmic functions.

The chain rule states that if we have a composite function f(g(x)), the derivative of f(g(x)) with respect to x is f'(g(x)) * g'(x).

Let's break down y = ln((3x + 2)^6(x + 10)^3):

Step 1: Write the function in exponential form:
(3x + 2)^6(x + 10)^3 = e^y

Step 2: Take the derivative of both sides with respect to x:
d/dx (e^y) = d/dx [(3x + 2)^6(x + 10)^3]

Step 3: Apply the chain rule and power rule:
d/dx (e^y) = e^y * [(6(3x + 2)^5)(x + 10)^3 + 3(3x + 2)^6(x + 10)^2]

Step 4: Simplify the expression:
dy/dx = e^y * [(6(3x + 2)^5)(x + 10)^3 + 3(3x + 2)^6(x + 10)^2]

Since y = ln((3x + 2)^6(x + 10)^3), we substitute e^y back into the expression:
dy/dx = [(3x + 2)^6(x + 10)^3] * [(6(3x + 2)^5)(x + 10)^3 + 3(3x + 2)^6(x + 10)^2]

Therefore, dy/dx = [(3x + 2)^6(x + 10)^3] * [(6(3x + 2)^5)(x + 10)^3 + 3(3x + 2)^6(x + 10)^2]