8 MORE RULESFORDERIVATIVESThe derivative of an inverse function 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):
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Problem 2. Use the chain rule to calculate the derivative of
See the Example, Lesson 6, and Lesson 22 of Algebra.
Proof of the quotient rule
Proof. Since g = g(x), then
according to the chain rule, and Problem 4 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 . 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'.
Problem 7. 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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