8 MORE RULES FOR DERIVATIVES The 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 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.
See the Example, Lesson 6.
Proof of the quotient rule
Proof. Since g = g(x), then
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.)
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
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? To see the answer, pass your mouse over the colored area. 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 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.
Problem 3. 15y + 5y3 + 3y5 = 15x. Calculate y'.
Problem 5. 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 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."
Therefore,
Next Lesson: Velocity and rates of change www.proyectosalonhogar.com |