Differentiate:
y = In ( (x+3) / (x^2) )
To differentiate the given function y = ln((x+3)/(x^2)), we can use the chain rule and the quotient rule.
Step 1: Rewrite the function using the properties of logarithms:
y = ln(x+3) - ln(x^2)
Step 2: Apply the chain rule to the first term:
dy/dx = (1/(x+3)) * d/dx(x+3) - d/dx(ln(x^2))
Step 3: Simplify the first term by taking the derivative of (x+3):
dy/dx = (1/(x+3)) * 1 - d/dx(ln(x^2))
Step 4: Simplify the second term by applying the derivative of the natural logarithm function:
dy/dx = 1/(x+3) - (1/x^2) * d/dx(x^2)
Step 5: Take the derivative of the second term:
dy/dx = 1/(x+3) - (1/x^2) * 2x
Step 6: Simplify further:
dy/dx = 1/(x+3) - 2/x
Thus, the derivative of y = ln((x+3)/(x^2)) is dy/dx = 1/(x+3) - 2/x.