# How are the derivatives of inverse functions related?

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science
space and astronomy

**Derivatives of inverse functions**.

**Functions**f and g are

**inverses**if f(g(x))=x=g(f(x)). For every pair of such

**functions**, the

**derivatives**f' and g' have a special relationship. Learn about this relationship and see how it applies to ??ˣ and ln(x) (which are

**inverse functions**!).

Then, what is the derivative of inverse functions?

The **Derivative** of an **Inverse Function**. (f−1)′(a)=pq. f′(f−1(a))=qp. (f−1)′(a)=1f′(f−1(a)).

**dy**/

**dx**, pronounced "dee y by dee x" .

Simply so, how do you do inverse functions?

**Finding the Inverse of a Function**

- First, replace f(x) with y .
- Replace every x with a y and replace every y with an x .
- Solve the equation from Step 2 for y .
- Replace y with f−1(x) f − 1 ( x ) .
- Verify your work by checking that (f∘f−1)(x)=x ( f ∘ f − 1 ) ( x ) = x and (f−1∘f)(x)=x ( f − 1 ∘ f ) ( x ) = x are both true.

The multiplicative **inverse of 6** is 1/**6**.