Use logarithmic differentiation to find the derivative of the function.
y = (tan x)^(7/x)
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To find the derivative of the function y = (tan x)^(7/x) using logarithmic differentiation, follow these steps:
Step 1: Take the natural logarithm of both sides of the equation:
ln(y) = ln((tan x)^(7/x))
Step 2: Use the logarithmic property of exponents to bring down the power as a coefficient:
ln(y) = (7/x) ln(tan x)
Step 3: Differentiate both sides of the equation implicitly with respect to x:
(1/y) * y' = (7/x) * ln(tan x) - (7/x^2) * ln(tan x) + (7/x) * (1/tan x) * sec^2(x)
Step 4: Simplify the expression:
y' = y * [(7/x) * ln(tan x) - (7/x^2) * ln(tan x) + (7/x) * (1/tan x) * sec^2(x)]
Step 5: Replace y with its original expression:
y' = (tan x)^(7/x) * [(7/x) * ln(tan x) - (7/x^2) * ln(tan x) + (7/x) * (1/tan x) * sec^2(x)]
Therefore, the derivative of the function y = (tan x)^(7/x) using logarithmic differentiation is y' = (tan x)^(7/x) * [(7/x) * ln(tan x) - (7/x^2) * ln(tan x) + (7/x) * (1/tan x) * sec^2(x)].