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

##### 1 Answer

#### Explanation:

The sum property for differentiating states:

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

If we let

Here, let

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,

*Note: You may have noticed that I wrote

We can now find the derivative of

If we let

let

By the chain rule:

*(Modified to use

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

Once again, by the sum rule:

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...

- Spilt up the function into two different functions
- Find the derivative using the appropriate differentiation technique
- Simplify and repeat for the other function