Question

(-x)^-1 derivation using first priciple​

Answer 1

You can easily use a more general expression for the derivative of f(x)=x−n .

So from the definition

f′(x)=limΔx→0(x+Δx)−n−x−nΔx

but

(x+Δx)−n=1(x+Δx)n

hence

1(x+Δx)n−1xn=xn−(x+Δx)n(x(x+Δx))n

so the limit becomes

limΔx→01Δxxn−(x+Δx)n(x(x+Δx))n

and developing the n-th power of the binomial we get

limΔx→01Δxxn−xn−nxn−1Δx−...(x(x+Δx))n=

−nxn−1x2n=−n1x2n+1−n=−nxn+1

as all the terms with higher power than 1 in Δx (ellipsis in the development of the binomial) tend to zero when Δx tend to zero.

So in the aforementioned case n=1 hence f′(x)=−1n2.

♡Maths Expert♡