How do you differentiate #f(x)=sqrt(1/x^2)# using the chain rule?

Answer 1

#d/dx sqrt(1/x^2) = - absx/x^3#

You can name:

#y(x) = 1/x^2 = x^-2#

so that:

#(df)/dx = (df)/(dy)dy/dx = d/dy(sqrt(y))d/dx(x^-2) = 1/(2sqrt(y))(-2x^-3) = 1/(2sqrt(1/x^2))(-2x^-3) = -sqrt(x^2)/x^3 = - abs(x)/x^3#

You can also note that:

#f(x) = sqrt(1/x^2) = 1/absx#

so that:

#{((df)/dx = d/dx (1/x) = -1/x^2" for " x > 0), ((df)/dx = d/dx (-1/x) = 1/x^2" for " x < 0):}#

which is clearly the same.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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