Reciprocal Rule in Differentiation

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

[1/g(x)]' = -g'(x)/[g(x)]2

Proof

Reciprocal Rule in Differentiation: Proof of the Formula

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

Find the derivative of the given function. y = 1/(x^3 + 2x)

y = 1/(x3 + 2x)

y' is equal to,
minus,
the derivative of x3 + 2x, (3x2 + 2⋅1x0)
over (x3 + 2x)2.

Derivatives of polynomials

So y' = -(3x2 + 2) / (x3 + 2x)2.