# How do you differentiate #f(x)=(x-3)^2+(-x-4)^2# using the sum rule?

We are basically differentiating twice if we split up the function into two separate functions:

It may be helpful to create a list that shows each function along with its derivative:

Following up on the chain rule, we can substitute each corresponding piece into the chain rule. Thus,

By the chain rule:

Now that we have found both derivatives separately we can combine them to answer the given question:

We know that:

Finally, simplify by distributing and combing like terms:

So our final answer can be expressed as:

It looks like a lot, but it really isn't, In essence when doing these types of problems...

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To differentiate ( f(x) = (x - 3)^2 + (-x - 4)^2 ) using the sum rule, first differentiate each term separately with respect to ( x ), and then add the results.

The derivative of ( (x - 3)^2 ) with respect to ( x ) is ( 2(x - 3) ).

The derivative of ( (-x - 4)^2 ) with respect to ( x ) is ( 2(-x - 4) ).

Therefore, the derivative of ( f(x) = (x - 3)^2 + (-x - 4)^2 ) with respect to ( x ) is ( 2(x - 3) + 2(-x - 4) ). Simplifying this expression gives ( 2x - 6 - 2x - 8 ), which further simplifies to ( -14 ). So, the derivative of ( f(x) ) with respect to ( x ) is ( -14 ).

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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.

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.

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.

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