求导：f(x) = (x/50)^(100/(50-x)) - 使用对数微分法

To find the derivative of the function f(x) = (x/50)^(100/(50-x)), we can use logarithmic differentiation.
Let y = (x/50)^(100/(50-x))
Taking the natural logarithm of both sides,
ln(y) = ln[(x/50)^(100/(50-x))]
Using the properties of logarithms, we can simplify this expression:
ln(y) = (100/(50-x)) * ln(x/50)
Now, we can differentiate both sides with respect to x:
(d/dx) ln(y) = (d/dx) [(100/(50-x)) * ln(x/50)]
Using the chain rule and product rule, we can differentiate the right side:
(d/dx) ln(y) = [(d/dx) (100/(50-x))] * ln(x/50) + (100/(50-x)) * (d/dx) ln(x/50)
Simplifying further:
(d/dx) ln(y) = [-100/(50-x)^2] * ln(x/50) + (100/(50-x)) * (d/dx) ln(x/50)
Now, let's find the derivative of ln(y):
(d/dx) ln(y) = (d/dx) ln[(x/50)^(100/(50-x))]
Using the chain rule, we have:
(d/dx) ln(y) = (d/dx) [(100/(50-x)) * ln(x/50)]
(d/dx) ln(y) = (100/(50-x)) * (d/dx) ln(x/50)
Substituting this back into the previous equation:
(100/(50-x)) * (d/dx) ln(x/50) = [-100/(50-x)^2] * ln(x/50) + (100/(50-x)) * (d/dx) ln(x/50)
We can cancel out the common factor of (100/(50-x)):
(d/dx) ln(x/50) = [-1/(50-x)^2] * ln(x/50) + (d/dx) ln(x/50)
Now, let's find the derivative of ln(x/50):
(d/dx) ln(x/50) = (d/dx) ln(x) - ln(50)
(d/dx) ln(x/50) = 1/x - ln(50)
Substituting this back into the equation:
1/x - ln(50) = [-1/(50-x)^2] * ln(x/50) + (d/dx) ln(x/50)
Now, we can solve for (d/dx) ln(x/50):
(d/dx) ln(x/50) = 1/x - ln(50) + [1/(50-x)^2] * ln(x/50)
Finally, to find the derivative of the original function f(x), we can multiply both sides by y = (x/50)^(100/(50-x)):
(d/dx) [y] = y * [(d/dx) ln(x/50)]
(d/dx) [y] = (x/50)^(100/(50-x)) * [1/x - ln(50) + [1/(50-x)^2] * ln(x/50)]
Therefore, the derivative of f(x) = (x/50)^(100/(50-x)) is:
(d/dx) [f(x)] = (x/50)^(100/(50-x)) * [1/x - ln(50) + [1/(50-x)^2] * ln(x/50)]