How do you find the logarithmic differentiation of y = x^6x?
You can use the same implicit differentiation technique that Reiny showed Ashley:
http://www.jiskha.com/display.cgi?id=1288893805
If you encounter problems, post what you've got.
To find the logarithmic differentiation of a function, follow these steps:
Step 1: Take the natural logarithm of both sides of the equation.
ln(y) = ln(x^6x)
Step 2: Apply logarithmic properties to simplify the equation.
ln(y) = (6x)ln(x)
Step 3: Differentiate both sides of the equation implicitly with respect to x.
(d/dx) ln(y) = (d/dx) [(6x)ln(x)]
Step 4: Use the chain rule on the right-hand side of the equation.
(d/dx) ln(y) = (d/dx) [6x * ln(x)] = 6 * ln(x) + 6x * (1/x)
Step 5: Simplify the right-hand side of the equation.
(d/dx) ln(y) = 6 * ln(x) + 6
Step 6: Multiply both sides of the equation by y to isolate dy/dx.
dy/dx * (1/y) = 6 * ln(x) + 6
Step 7: Rewrite y as x^6x.
dy/dx * (1/x^6x) = 6 * ln(x) + 6
Step 8: Multiply both sides of the equation by x^6x to solve for dy/dx.
dy/dx = (6 * ln(x) + 6) * x^6x
Therefore, the derivative of y = x^6x with logarithmic differentiation is dy/dx = (6 * ln(x) + 6) * x^6x.