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If #f(x)= sin(- x -1) # and #g(x) = 4x^2 -5 #, how do you differentiate #f(g(x)) # using the chain rule?

Answer 1

William Abdalla

#d/dx(f[g(x)])=-8xcos(5-4x^2)#

#f[g(x)]=f(4x^2-5)#

#=sin[-(4x^2-5)]#

#=sin(5-4x^2)#

#therefore d/dx[sin(5-4x^2)]=-8xcos(5-4x^2)#

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Answer 2

Emilia Aguilar

To differentiate ( f(g(x)) ) using the chain rule, follow these steps:

Find ( f'(x) ) and ( g'(x) ).
Substitute ( g(x) ) into ( f(x) ) to get ( f(g(x)) ).
Differentiate ( f(g(x)) ) with respect to ( x ) using the chain rule.

Given: ( f(x) = \sin(-x - 1) ) ( g(x) = 4x^2 - 5 )

Calculate ( f'(x) ) and ( g'(x) ): ( f'(x) = -\cos(-x - 1) ) ( g'(x) = 8x )
Substitute ( g(x) ) into ( f(x) ) to get ( f(g(x)) ): ( f(g(x)) = \sin(-(4x^2 - 5) - 1) )
Differentiate ( f(g(x)) ) with respect to ( x ) using the chain rule: ( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) ) ( \frac{d}{dx}[f(g(x))] = -\cos(-(4x^2 - 5) - 1) \cdot 8x )

Therefore, the derivative of ( f(g(x)) ) with respect to ( x ) using the chain rule is ( -8x\cos(4x^2 - 6) ).

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Answer from HIX Tutor

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