What is derivative of the function f(x)=(x^2-4)^(1/(x-2))?

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We have to differentiate f(x)=(x^2-4)^(1/(x-2))

We can make the differentiation easier by eliminating the exponent on the right. For this, we take the log to the base e of both the sides.

ln f(x) = ln[(x^2-4)^(1/(x-2))]

=> ln f(x) = (1/(x - 2))*ln (x^2 - 4)

differentiate both the sides

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We have to differentiate f(x)=(x^2-4)^(1/(x-2))

We can make the differentiation easier by eliminating the exponent on the right. For this, we take the log to the base e of both the sides.

ln f(x) = ln[(x^2-4)^(1/(x-2))]

=> ln f(x) = (1/(x - 2))*ln (x^2 - 4)

differentiate both the sides

f'(x)/f(x) = -1*(1/(x - 2)^2)* ln(x^2 - 4) + (1/(x - 2))*2x*(1/(x^2 - 4)

f'(x) = f(x)*[- ln(x^2 - 4)/(x - 2)^2 + 2x/(x - 2)*(x^2 - 4)

The required result is f(x)*[-ln(x^2 - 4)/(x - 2)^2 + 2x/(x - 2)*(x^2 - 4)]

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