How do you differentiate #f(x)=sinx+cosx-x^3# using the sum rule?

Answer 1

#f'(x)=cosx-sinx-3x^2#

The sum rule basically states that to find the derivative of a sum, you can take the derivative of each individual part and add them together to find the derivative of the entire function.

In other words:

#d/dx(u+v+w...)=(du)/dx+(dv)/dx+(dw)/dx+...#

Thus,

#f'(x)=color(red)(d/dx[sinx])+color(blue)(d/dx[cosx])+color(green)(d/dx[-x^3]#

#f'(x)=color(red)(cosx)+color(blue)(-sinx)+color(green)(-3x^2)#

#f'(x)=cosx-sinx-3x^2#

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Answer 2

Adam Adkins

To differentiate the function ( f(x) = \sin(x) + \cos(x) - x^3 ) using the sum rule, you differentiate each term individually and then add them together. The derivative of ( \sin(x) ) is ( \cos(x) ), the derivative of ( \cos(x) ) is ( -\sin(x) ), and the derivative of ( x^3 ) is ( 3x^2 ). Therefore, the derivative of ( f(x) ) is ( \cos(x) - \sin(x) - 3x^2 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.