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.
Problem 2. Use the chain rule to calculate the derivative of
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.)
Let us calculate .
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. As for y2, we consider it implicitly a function of x, and therefore we may apply the chain rule to it. Then we will solve for .
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."
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