Calculus
posted by Lauren .
I have a two part question that pertains to a curve (r(x)) and its tangent line at x=3.
We are given that at x=3, r(x)=8. In order to find the slope of the tangent line, we are given another point (on the tangent line): (3.2, 8.5). Therefore the slope of the tangent line at x=3 is 5/2
A. Let c(x)= 1/r(x). Find c'(3).
I used the quotient rule (for clarity, I am substituting "a" for r(x))
c'(x)= (a[0]  1a')/a^2
Which reduces to
a'/a^2
Replacing r(x) at x=3 for a, we get:
r'(3)/(r(3))^2
Since the derivative of r(x) at 3 is equal to the slope of the tangent line at x=3, we get
(5/2)/(8)^2 or 5/128
Does that look correct?
b. Let i(x) = the inverse of r(x). Find i'(8).
I had some trouble with this one. I know that the point (3,8) is on r(x) (from the given information in part a). This would mean that if i(x) is the inverse of r(x), then (8,3) would be on i(x). I also know that r'(3)=5/2. Because derivatives of inverses are reciprocals at corresponding points, would this mean that i'(8)=2/5?

Calculus 
Reiny
I agree with your thinking for both questions.
I especially like that you realize the relationship of slopes between corresponding points on a relation and its inverse.
Respond to this Question
Similar Questions

Calculus  Tangent Line
Hi, im having problems with the following problem. The main issue is actually starting the problem. Find the two points on the curve y = x^4  2x^2  x that have a common tangent line. First, find the derivative of y(x) so that you … 
Calculus
Find an equation of the tangent line to the curve at the given point. (Assume the tangent line form to be f(x) = mx + b.) y= e^x/x (1,e) I got m= 0 nd b=e could anyone tell if its right? 
College Calculus
Let f(x) = 3x^2  2x + 1 a) write an equation of the line going through the points (1, f(1)), and (2, f(2)). b) Find a point on the graph of f where the tangent line to the graph has the same slope as the line in part (a). Write the … 
Applied Calculus......
Find the slope m of the tangent line to the graph of the function at the given point and determine an equation of the tangent line. f(x)=6x5x^2 at (2, 32) I have to use the four step process, but seem to be getting stuck... 
Calculus
Find the slope m of the tangent line to the graph of the function at the given point and determine an equation of the tangent line. f(x) = 15/(4 x) at (1,(15/4) m = y = 
Calculus
Consider the curve given by y^2 = 2+xy (a) show that dy/dx= y/(2yx) (b) Find all points (x,y) on the curve where the line tangent to the curve has slope 1/2. (c) Show that there are now points (x,y) on the curve where the line tangent … 
calculus
find an equation of the tangent line to the curve at the given point. graph the curve and the tangent line y=x/x1 at (2,2) ) 
calculus
Find the slope of the tangent line to the curve (sqrt(4x+y))+(sqrt(3xy)) at the point (3,4) The slope of the tangent line to the curve at the given point is ? 
Calculus AB
Sorry but I've got a lot of problems that I don't understand. 1) Let f(x)= (3x1)e^x. For which value of x is the slope of the tangent line to f positive? 
Calculus AB
Could someone please help me with these tangent line problems?