# Reciprocal Rule in Differentiation

How to solve the reciprocal rule in differentiation problems: formula, proof, example, and its solution.

## Formula

[1/*g*(*x*)]' = -*g*'(*x*)/[*g*(*x*)]^{2}

## Proof

Definition of a derivative function

Write the denominator *h* at the front part. (= 1/*h*)

Solve 1/*g*(*x* + *h*) - 1/*g*(*x*).

Adding and subtracting rational expressions

Write (-) sign in front of 1/*h*.

And, to undo the (-) sign,

change *g*(*x*) - *g*(*x* + *h*)

to *g*(*x* + *h*) - *g*(*x*).

Combine 1/*h* and [*g*(*x* + *h*) - *g*(*x*)].

Then the remaining part is 1/[*g*(*x* + *h*)*g*(*x*)].

The limit of the gray part is *g*'(*x*).

And the limit of the latter part is 1/[*g*(*x* + 0)*g*(*x*)].

So [1/*g*(*x*)]' = -*g*'(*x*)/[*g*(*x*)]^{2}.

## Example

*y* = 1/(*x*^{3} + 2*x*)*y*' is equal to,

minus,

the derivative of *x*^{3} + 2*x*, (3*x*^{2} + 2⋅1*x*^{0})

over (*x*^{3} + 2*x*)^{2}.

Derivatives of polynomials

So *y*' = -(3*x*^{2} + 2) / (*x*^{3} + 2*x*)^{2}.