What is the derivative of 1/(sqrt(x^4+x^2+1)) ?

The derivative of 1/(sqrt(x^4+x^2+1)) is -(x(2x^2+1))/((x^4+x^2+1)^{3/2)}

What is the first derivative of 1/(sqrt(x^4+x^2+1)) ?

The first derivative of 1/(sqrt(x^4+x^2+1)) is -(x(2x^2+1))/((x^4+x^2+1)^{3/2)}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

\frac{d}{d x} (\frac{1}{x ^{4} + x ^{2} + 1})

- \frac{x ( 2 x ^{2} + 1 )}{( x ^{4} + x ^{2} + 1 ) ^{\frac{3}{2}}}

Solution steps

\frac{d}{d x} (\frac{1}{x ^{4} + x ^{2} + 1})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= \frac{d}{d x} ((x^{4} + x^{2} + 1)^{- \frac{1}{2}})

Apply the chain rule:

- \frac{1}{2 ( x ^{4} + x ^{2} + 1 ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{4} + x^{2} + 1)

= - \frac{1}{2 ( x ^{4} + x ^{2} + 1 ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{4} + x^{2} + 1)

\frac{d}{d x} (x^{4} + x^{2} + 1) = 4 x^{3} + 2 x

= - \frac{1}{2 ( x ^{4} + x ^{2} + 1 ) ^{\frac{3}{2}}} (4 x^{3} + 2 x)

Simplify

- \frac{1}{2 ( x ^{4} + x ^{2} + 1 ) ^{\frac{3}{2}}} (4 x^{3} + 2 x) : - \frac{x ( 2 x ^{2} + 1 )}{( x ^{4} + x ^{2} + 1 ) ^{\frac{3}{2}}}

= - \frac{x ( 2 x ^{2} + 1 )}{( x ^{4} + x ^{2} + 1 ) ^{\frac{3}{2}}}

What is the derivative of 1/(sqrt(x^4+x^2+1)) ?
The derivative of 1/(sqrt(x^4+x^2+1)) is -(x(2x^2+1))/((x^4+x^2+1)^{3/2)}
What is the first derivative of 1/(sqrt(x^4+x^2+1)) ?
The first derivative of 1/(sqrt(x^4+x^2+1)) is -(x(2x^2+1))/((x^4+x^2+1)^{3/2)}