# How do you differentiate #f(x)=1/(x-1)^2+x^3-4/x# using the sum rule?

##### 1 Answer

Jan 22, 2016

#### Explanation:

The sum rule states that we can find the derivative of each added part individually and then combine them, like so:

#f'(x)=d/dx[1/(x-1)^2]+d/dx[x^3]-d/dx[4/x]#

Finding the derivative of the first term: we can redefine the function using negative exponents, and then differentiate using the chain rule.

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

#=-2(x-1)^-3(1)=color(red)(-2/(x-1)^3#

The second term will require a simple application of the chain rule.

#d/dx[x^3]=color(red)(3x^2#

The third can also be rewritten with negative exponents.

#d/dx[4x^-1]=-1(4x^-2)=color(red)(-4/x^2#

Thus, the derivative of the function is

#f'(x)=-2/(x-1)^3+3x^2-(-4/x^2)#

Slightly simplified/reordered:

#f'(x)=3x^2+4/x^2-2/(x-1)^3#