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Popular >

derivative of y=(x+1/x)^{-1}

Solution

derivative of

y = (x + \frac{1}{x})^{- 1}

Solution

- \frac{x ^{2} - 1}{( x ^{2} + 1 ) ^{2}}

Solution steps

\frac{d}{d x} ((x + \frac{1}{x})^{- 1})

Apply the chain rule:

- \frac{1}{( x + \frac{1}{x} ) ^{2}} \frac{d}{d x} (x + \frac{1}{x})

= - \frac{1}{( x + \frac{1}{x} ) ^{2}} \frac{d}{d x} (x + \frac{1}{x})

\frac{d}{d x} (x + \frac{1}{x}) = 1 - \frac{1}{x ^{2}}

= - \frac{1}{( x + \frac{1}{x} ) ^{2}} (1 - \frac{1}{x ^{2}})

Simplify

- \frac{1}{( x + \frac{1}{x} ) ^{2}} (1 - \frac{1}{x ^{2}}) : - \frac{x ^{2} - 1}{( x ^{2} + 1 ) ^{2}}

= - \frac{x ^{2} - 1}{( x ^{2} + 1 ) ^{2}}

Popular Examples

integral of (cos(t))/(4+sin(t))

\int \frac{cos ( t )}{4 + sin ( t )} d t

derivative of g(x)=3(4-9x)^4 derivative of

g (x) = 3 (4 - 9 x)^{4}

integral of (2e^x)/(e^{2x)+8e^x+16}

\int \frac{2 e ^{x}}{e ^{2 x} + 8 e ^{x} + 16} d x

limit as x approaches-3+of 1/(x+3)

x \to - 3 + lim (\frac{1}{x + 3})

tangent of f(x)=-x^3+4x^2,\at x=3 tangent of

f (x) = - x^{3} + 4 x^{2}, at x = 3

Frequently Asked Questions (FAQ)

What is the derivative of y=(x+1/x)^{-1} ?
The derivative of y=(x+1/x)^{-1} is -(x^2-1)/((x^2+1)^2)
What is the first derivative of y=(x+1/x)^{-1} ?
The first derivative of y=(x+1/x)^{-1} is -(x^2-1)/((x^2+1)^2)