How do you differentiate #f(x)=(x-5)^3-x^2+5x# using the sum rule?

Answer 1

#f'(x)=3x^2-32x+30#

The sum rule is simple. All we have to do is find the derivative of each part of the sum and add them back to one another.

Therefore, #f'(x)=stackrel"chain rule"overbrace(d/dx[(x-5)^3])-stackrel"nx^(n-1)"overbrace(d/dx[x^2])+stackrel"nx^(n-1)"overbrace(d/dx[5x])#

I've written the rules we'll need to continue in finding the derivatives.

Through the Chain Rule:

#d/dx[(x-5)^3]=3(x-5)^2*d/dx[x]=3(x-5)^2*1=3(x-5)^2#

#d/dx[x^2]=2x#

#d/dx[5x]=5#

We can add all these back together:

#f'(x)=3(x-5)^2-2x+5#

And, simplify:

#f'(x)=3(x^2-10x+25)-2x+5#

#f'(x)=3x^2-30x+25-2x+5#

#f'(x)=3x^2-32x+30#

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Answer 2

Adam Adkins

To differentiate the given function ( f(x) = (x - 5)^3 - x^2 + 5x ) using the sum rule, differentiate each term separately and then add them together. The sum rule states that the derivative of a sum is the sum of the derivatives of the individual terms.

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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.