*Notes to a video lecture on http://www.unizor.com*

__Derivative Properties -__

Reciprocal to a Function

Reciprocal to a Function

Our purpose is to express the derivative of a reciprocal of a function in terms of its own derivative.

Assume that the real function

**is defined and**

*f(x)**differentiable*(that is, its derivative exist) on certain interval with a derivative

*f**.*

^{ I}**(x)**Let's determine the derivative of its reciprocal

**wherever**

*h(x) = 1/f(x)***.**

*f(x) ≠ 0*The increment of function

**is**

*h(x)*Δ

**=**

*h(x)***Δ**

*h(x+***Δ**

*x)−h(x) =*

= 1/f(x+= 1/f(x+

**Δ**

*x) − 1/f(x) =*

= [f(x)−f(x+= [f(x)−f(x+

**Δ**

*x)]/[f(x+***Δ**

*x)·f(x)]*

= −= −

**Δ**

*f(x)/[f(x+*

*x)·f(x)]*Next operations to find a derivative are: dividing the function increment Δ

**by an increment of an argument Δ**

*h(x)***and going to a limit as Δ**

*x***.**

*x→0*Δ

**Δ**

*h(x)/***Δ**

*x =*

= [−= [−

**Δ**

*f(x)/***Δ**

*x)]/[f(x+*

*x)·f(x)]*Since function

**is differentiable, there is a limit of Δ**

*f(x)***Δ**

*f(x)/***as Δ**

*x***. This limit is**

*x→0*

*f**.*

^{ I}**(x)**At the same time

**Δ**

*f(x+*

*x) → f(x)*as Δ

**.**

*x→0*Recall the properties of limits:

if two sequences have limits

*L*and

*M*, then their product has a limit

*L·M*;

if two sequences have limits

*L*and

*M ≠ 0*, then their ratio has a limit

*L/M*.

Therefore,

Δ

**Δ**

*h(x)/***→**

*x*

*−f*

^{ I}**(x)/f²(x)**Hence,

*h*

^{I}**(x) = −f**^{ I}**(x)/f²(x)**It's called the Product Rule of differentiation.

Different forms of notation of this rule are:

(1)

*[1/f(x)]*

^{ I}**= −f**^{ I}**(x)/f²(x)**(2)

*d*

**[1/f(x)]**/dx =**−[**d**f(x)**/dx**]·[1/f²(x)]***Examples*

*[sec(x)]*

^{ I}**= [1/cos(x)]**^{ I}**=**

= −[cos(x)]= −[cos(x)]

^{ I}**/cos²(x) =**

= sin(x)/cos²(x)= sin(x)/cos²(x)

*(d/dx)*

**[tan(x)] =**

=(d/dx)=

**[sin(x)/cos(x)] =**

=(d/dx)=

**{[sin(x)]·[1/cos(x)]} =**

=(d/dx)=

**[sin(x)]·[1/cos(x)] + sin(x)·**(d/dx)**[1/cos(x)] =**

= cos(x)·[1/cos(x)] + sin(x)·sin(x)/cos²(x) =

= 1+tan²(x)= cos(x)·[1/cos(x)] + sin(x)·sin(x)/cos²(x) =

= 1+tan²(x)

*d***e**/dx^{−x}**=**

=(d/dx)=

**(1/e**

= −[(d/dx)^{x}) == −[

**(e**

= −e

= −1/e

= −e^{x})]/[(e^{x})²] == −e

^{x}/[(e^{x})²] == −1/e

^{x}== −e

^{−x}*D*_{x}**[f(x)/g(x)] =**

= [D= [

_{x}**f(x)]·[1/g(x)] + f(x)·{**D_{x}**[1/g(x)]} =**

= [D= [

_{x}**f(x)]·[1/g(x)] − f(x)·[**D_{x}**g(x)]/g²(x) =**

= g(x)·[D= g(x)·[

_{x}**f(x)]·/g²(x) − f(x)·[**D_{x}**g(x)]/g²(x) =**

= [g(x)·f= [g(x)·f

^{ I}**(x)−f(x)·g**^{ I}**(x)]/g²(x)**
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