Find the first and second derivatives of the functions u=(x2+x)(x2−x+1)x4

Given: The given function is u=(x2+x)(x2−x+1)x4 find the first derivative: dudx=ddx[(x2+x)(x2−x+1)x4] =ddx[(x4+x3−x3−x2+x2+x)x4] =ddx[(x4+x)x4] on further simplification, dudx=(x4)(4x3+1)−(x4+x)(4x3)(x4)2 =4x7+x4−4x7−4x4x8 =−3x4x8 =−3x4 find the second derivative: d2udx2=ddx(−3x4) =−3ddx(x−4) =12x−5 =12x5

Key to "Find the first and second derivatives of the..."

Alfred Martin

Answered question

2021-12-08

Find the first and second derivatives of the functions

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

Answer & Explanation

sonorous9n

Beginner2021-12-09Added 34 answers

Given:
The given function is

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

find the first derivative:

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

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

= \frac{d}{d x} [\frac{(x^{4} + x)}{x^{4}}]

on further simplification,

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

= \frac{4 x^{7} + x^{4} - 4 x^{7} - 4 x^{4}}{x^{8}}

= \frac{- 3 x^{4}}{x^{8}}

= - \frac{3}{x^{4}}

find the second derivative:

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

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

= 12 x^{- 5}

= \frac{12}{x^{5}}

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