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How do you use the chain rule to differentiate #f(x)=sin(1/(x^2+1))#?

Answer 1

Luke Alvarez

See below

#f(x)=Sin(1/(x^2+1))#

#f(x)=Sin(u)#

#f'(x)=Cos(u) times u'#

#u=1/(x^2+1)=(x^2+1)^-1#

(Substitute again to include the inner function)

#v=x^2+1# #v'=2x#

#u'=-1(v')(v)^-2=-1(2x)(x^2+1)^-2=(-2x)/(x^2+1)^2#

Hence:

#f'(x)=((-2x)/(x^2+1)^2)Cos(1/(x^2+1))#

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Answer 2

Cameron Alexander

To differentiate ( f(x) = \sin\left(\frac{1}{x^2 + 1}\right) ) using the chain rule, follow these steps:

Let ( u = \frac{1}{x^2 + 1} ).
Find ( \frac{du}{dx} ).
Differentiate ( \sin(u) ) with respect to ( u ), which is ( \cos(u) ).
Multiply ( \frac{du}{dx} ) by ( \cos(u) ).

The derivative of ( f(x) ) is given by ( f'(x) = \cos\left(\frac{1}{x^2 + 1}\right) \times \frac{-2x}{(x^2 + 1)^2} ).

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Answer from HIX Tutor

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