y=ln(x+1) +ln(x-1) find dy/dx
I have trouble with this problem since i'm not that familar with finding the derivative when they involve ln. Please show me how to solve this problem. I should have gotten 2x/(x^2-1)
To find the derivative of y = ln(x+1) + ln(x-1), we can use the logarithmic differentiation method.
Here's how to solve it step by step:
Step 1: Start with the given function y = ln(x+1) + ln(x-1).
Step 2: Apply the logarithmic differentiation technique, which involves taking the natural logarithm of both sides of the equation:
ln(y) = ln[ln(x+1) + ln(x-1)]
Step 3: Simplify the right side of the equation using the logarithmic properties. When taking the natural logarithm of the sum of two terms, we can express it as the sum of the natural logarithms of each term:
ln(y) = ln(x+1) + ln(x-1)
Step 4: Differentiate both sides of the equation with respect to x. Remember that the derivative of ln(x) is 1/x, and the chain rule applies here:
(d/dx)ln(y) = (d/dx)ln(x+1) + (d/dx)ln(x-1)
(dy/y) = (1/(x+1))*(d/dx)(x+1) + (1/(x-1))*(d/dx)(x-1)
dy/y = (1/(x+1))*1 + (1/(x-1))*1
Step 5: Solve for dy/dx by multiplying both sides of the equation by y:
dy = y[(1/(x+1)) + (1/(x-1))]
Step 6: Replace y with its original expression: ln(x+1) + ln(x-1):
dy = [ln(x+1) + ln(x-1)][(1/(x+1)) + (1/(x-1))]
Step 7: Simplify the expression inside the brackets by finding a common denominator:
dy = [ln(x+1)*(x-1) + ln(x-1)*(x+1)] / [(x+1)(x-1)]
Step 8: Expand the expression inside the brackets:
dy = [(xln(x+1) - ln(x+1) + xln(x-1) - ln(x-1))] / [(x+1)(x-1)]
Step 9: Combine like terms in the numerator:
dy = [xln(x+1) + xln(x-1) - ln(x+1) - ln(x-1)] / [(x+1)(x-1)]
Step 10: Simplify the denominator:
dy = [xln(x+1) + xln(x-1) - ln(x+1) - ln(x-1)] / (x^2 - 1)
And there you have it. The derivative dy/dx of y = ln(x+1) + ln(x-1) is:
dy/dx = [xln(x+1) + xln(x-1) - ln(x+1) - ln(x-1)] / (x^2 - 1)
Which can be further simplified to:
dy/dx = [(xln(x+1) - ln(x+1)) + (xln(x-1) - ln(x-1))] / (x^2 - 1)