What is the derivative of 1/sqrt(1 - x^2)1√1−x2?

2 Answers
Ada C.
Apr 12, 2018

=(x)(1-x^2)^(-3/2)=(x)(1−x2)−32

Explanation:

f(x)=1/sqrt(1-x^2)=(sqrt(1-x^2))^-1=(1-x^2)^(-1/2)f(x)=1√1−x2=(√1−x2)−1=(1−x2)−12
d/dx f(x)=-1/2(1-x^2)^(-1/2-1)(0-2x)ddxf(x)=−12(1−x2)−12−1(0−2x)
=(2x)/2(1-x^2)^(-3/2)=2x2(1−x2)−32
=(x)(1-x^2)^(-3/2)=(x)(1−x2)−32

Aviv S.
Apr 12, 2018

The derivative is (xsqrt(1-x^2))/(1-x^2)^2x√1−x2(1−x2)2.

Explanation:

Using the quotient rule:

color(white)=d/dx[1/sqrt(1-x^2)]=ddx[1√1−x2]

=(d/dx[1]*sqrt(1-x^2)-1*d/dx[sqrt(1-x^2)])/(sqrt(1-x^2))^2=ddx[1]⋅√1−x2−1⋅ddx[√1−x2](√1−x2)2

=(0*sqrt(1-x^2)-1*d/dx[sqrt(1-x^2)])/(1-x^2)=0⋅√1−x2−1⋅ddx[√1−x2]1−x2

=(-d/dx[sqrt(1-x^2)])/(1-x^2)=−ddx[√1−x2]1−x2

Chain rule:

=(-1/(2sqrt(1-x^2))*d/dx[1-x^2])/(1-x^2)=−12√1−x2⋅ddx[1−x2]1−x2

=(-1/(2sqrt(1-x^2))*-2x)/(1-x^2)=−12√1−x2⋅−2x1−x2

=(x/sqrt(1-x^2))/(1-x^2)=x√1−x21−x2

=((xsqrt(1-x^2))/(1-x^2))/(1-x^2)=x√1−x21−x21−x2

=(xsqrt(1-x^2))/(1-x^2)^2=x√1−x2(1−x2)2

That's the derivative. Hope this helped!

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