Question: "Find the derivative of the following functions f(x)=(1+1x2)(x2+1)"

ofodse

Answered question

2021-12-04

Find the derivative of the following functions

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

Mary Ramirez

Beginner2021-12-05Added 19 answers

Step 1
Derivative of

x^{n} i s n x^{n - 1} f or n \neq 0

. Derivative of a constant is equal to 0. One property of derivatives is

\frac{d}{d x} (f (x) + g (x)) = \frac{d}{d x} f (x) + \frac{d}{d x} g (x)

; i.e., derivative of sum of functions is the sum of derivatives of the two functions.
The product rule of differentiation states that

\frac{d}{d x} (f (x) g (x)) = \frac{d}{d x} (f (x)) g (x) + f (x) \frac{d}{d x} (g (x))

. Ue the product rule and the rule for derivative of sum of functions to find the derivative of the given function.
Step 2
Given function is

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

. Use the product rule of differentiation to compute the first derivative.

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

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

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

= (0 - \frac{2}{x^{3}}) (x^{2} + 1) + (1 + \frac{1}{x^{2}}) (2 x + 0)

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

= - \frac{2}{x} - \frac{2}{x^{3}} + 2 x + \frac{2}{x}

= - \frac{2}{x^{3}} + <

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