Get the solution to "Find the derivatives of f(x)=xcos⁡x1+x2" - Plainmath

waigaK

Answered question

2021-09-29

Find the derivatives of

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

Answer & Explanation

sweererlirumeX

Skilled2021-09-30Added 91 answers

Step 1
Given:

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

Step 2
Differentiating wrt "x"

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

f (x) = \frac{\sqrt{1 + x^{2}} (- x \sin x + \cos x) - x \cos x \frac{1}{2 \sqrt{1 + x^{2}}} (2 x)}{(1 + x^{2})}

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

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

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

= {(1 + x^{2})}^{- \frac{3}{2}} [\cos x - x (1 + x^{2}) \sin x]

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