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
For example, accepting for the moment that the derivative of sin x is cos x (Lesson 12):
To see the answer, pass your mouse over the colored area.
Proof of the quotient rule
Proof. Since g = g(x), then
Therefore, according to the product rule (Lesson 6),
This is the quotient rule, which we wanted to prove.
Consider the following:
x2 + y2 = r2
This is the equation of a circle with radius r. (Lesson 17 of Precalculus.)
derivative. But rather than do that, we will take the derivative of each term. As for y2, we consider it implicitly a function of x, and therefore
This is called implicit differentiation. We treat y as a function of x and apply the chain rule. The derivative that results generally contains both x and y.
Problem 5. 15y + 5y3 + 3y5 = 5x3. Calculate y'.
a) In this circle,
x2 + y2 = 25,
a) what is the y-coördinate when x = −3?
y = 4 or −4. For,
(−3)2 + (±4)2 = 52
b) What is the slope of the tangent to the circle at (−3, 4)?
c) What is the slope of the tangent to the circle at (−3, −4)?
Problem 8. 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.
Problem 9. Calculate the slope of the tangent to this curve at (2, −1):
x3 − 3xy2 + y3 = 1
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
Let us write
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:
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 following:
Theorem. If g(x) is the inverse of f(x), then
"The derivative of an inverse function is equal to
the reciprocal of the derivative of the direct function
when its argument is the inverse function."
Please make a donation to keep TheMathPage online.
Copyright © 2013 Lawrence Spector
Questions or comments?