Question

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

Answer 1

You can apply the sum rule right away to the two expressions added.

`d/dx[(x-3)^2+(x-4)^3)]=d/dx[(x-3)^2]+d/dx[(x-4)^3]`

You can then differentiate each part using the chain rule.

`=2(x-3)d/dx(x-3)+3(x-4)^2d/dx(x-4)`

`=2(x-3)(1)+3(x-4)^2(1)" "`The derivative terms go to `1`

`=2x-6+3(x^2-8x+16)" "`Expand the squared term

`=2x-6+3x^2-24x+48" "`Multiply the `3` through

Combining like terms, we get

`=3x^2-22x+42`