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8

MORE RULES

FOR

DERIVATIVES

The quotient rule


The quotient rule

The following is called the quotient rule:

"The derivative of the quotient of two functions

is equal to
the denominator times the derivative of the numerator
minus the numerator times the derivative of the denominator
all divided by the square of the denominator."

For example, accepting for the moment that the derivative of sin x is cos x (Lesson 12):

   Problem 1.   Calculate the derivative of      x²  
sin x
.

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sin x· 2xx² cos x
         sin2x
   Problem 2.   Use the chain rule to calculate the derivative of   sin²x
  x3
.
x³· 2 sin x cos x − sin²x· 3x²
                 x6
  =   x² sin x(2x cos x − 3 sin x)
                x6
 
    =   sin x(2x cos x − 3 sin x)
               x4
   Problem 3.   Calculate the derivative of    x² − 5x − 6
    2x + 1
.
(2x + 1)(2x − 5) − (x² − 5x − 6)· 2
                  (2x + 1)²
 =  4x² − 8x − 5 − 2x² + 10x + 12
                  (2x + 1)²
 
   =  2x² + 2x + 7
   (2x + 1)²
   Problem 4.   Calculate the derivative of    3x² − x + 4
     
.

See the Example, Lesson 6.

 = 
 
   = 
 
   = 

Proof of the quotient rule

THEOREM.

Proof.   Since g = g(x), then

 d 
dx
1
g
 =    d 
dg
  1
g
·   dg 
dx
 =  − 1
g²
 g'

according to the chain rule, and Problem 3 of Lesson 5.

Therefore, according to the product rule (Lesson 6),

This is the quotient rule, which we wanted to prove.



Implicit differentiation

Consider the following:

x² + y² = r²

This is the equation of a circle with radius r. (Lesson 17 of Precalculus.)

  Let us calculate    dy
dx
.   To do that, we could solve for y and then take the

derivative.  But rather than do that, we will take the derivative of each term.  We will assume that y is a function of x, and we will apply the

  chain rule.  Then we will solve for  dy
dx
.

 d 
dx
  x²   +    d 
dx
  y²   =    d
dx
  r²
2x + 2y   dy 
dx
  =   0
 
dy 
dx
  =   x
y
.

This is called implicit differentiation.  y is implicitly a function of x.  The result generally contains both x and y.

Problem 1.

a)  In this circle,

x² + y² = 25,

a)   what is the y-coordinate when x = −3?

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y = 4 or −4. For,

(−3)² + (±4)² = 5²

b)  What is the slope of the tangent to the circle at (−3, 4)?

3
4
.  For, the derivative is − x
y
.

c)  What is the slope of the tangent to the circle at (−3, −4)?

3
4

Problem 2.   In the first quadrant, what is the slope of the tangent to this circle,

(x − 1)² + (y + 2)² = 169,

when x = 6?

[Hint:  5² + 12² = 13² is a Pythagorean triple.]

In the first quadrant, when x = 6, y = 10.

(6 − 1)² + (10 + 2)² = 13².

y' = − x − 1
y + 2
.  Therefore the slope is −   6 − 1
10 + 2
  =  −  5 
12

Problem 3.   15y + 5y3 + 3y5 = 15x.  Calculate y'.

15y' + 15y²y' + 15y4y' = 15
 
y'(1 + y² + y4) = 1
 
y' =        1       
1 + y² + y4
   Problem 4.      Calculate y'.
= 0
 
=
 
y' =

Problem 5.   Calculate the slope of the tangent to this curve at (2, −1):

x3 − 3xy² + y3 = 1

  3x² − (3x· 2y y' + y²· 3) + 3y² y'  =  0
 
according to the product rule.
 
  3x² − 6xy y' − 3y² + 3y² y'  =  0
 
  x² − 2xy y'y² + y² y'  =  0
 
  y'(y² − 2xy)  =  y² − x²
 
  y'  =   y² − x² 
y² − 2xy
 
 Therefore, at (2, −1):
 
  y'  =      (−1)² − 2²    
(−1)² − 2· 2· −1
 
     =  −3
  5
 
     =  3
5

The derivative of an inverse function

When we have a function  y = f(x) -- for example

y = x²

-- then we can often solve for x.  In this case,

On exchanging the variables, we have

is called the inverse function of  yx².

Let us write

f(x) = x²
 
g(x) =

And let us call f the direct function and g the inverse function.  The formal relationship between f and g is the following:

f( g(x)) = g( f(x)) = x.

(Topic 19 of Precalculus.)

Here are other pairs of direct and inverse functions:

f(x) = sin x   g(x) = arcsin x
 
f(x) = ax   g(x) = logax
 
f(x) = x3   g(x) =

Now, when we know the derivative of the direct function f, then from it we can determine the derivative of g.

Thus, let g(x) be the inverse of f(x).  Then

f(g(x) = x.

Now take the derivative with respect to x:

This implies:


"The derivative of an inverse function is equal to

the reciprocal of the derivative of the direct function

when it is expressed as a function of the inverse function."

   Example.   Let f(x) = x²,  and    Then  f( g) = g².  

Therefore,

Next Lesson:  Velocity and rates of change


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