Derivative and Tangent Line Equation

In summary, the conversation discussed the process of finding the equation of a tangent line to a graph at a specific point. The participants used the quotient rule to find the derivative, but there was some confusion about the sign of an exponent. They also clarified the difference between a line being tangent to a function and crossing it at multiple points.
  • #1
physstudent1
270
1
[SOLVED] Derivative and tangent line

Homework Statement


y = (x)/(x+x^-1)

Find an equation of the tanget line to the graph at x=2


Homework Equations





The Attempt at a Solution



I used the quotient rule and got the derivative to be 0 / (x+x^-1) = y' so then the slope is 0 of the tangent line; and I plugged x=2 into the original equation to find the corresponding y value which was 4/5 so my equation is just y=4/5 ? is this correct it seems odd and looks odd on my calculator
 
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  • #2
How did you get 0 / (x+x^-1) = y' ?
 
  • #3
I got [(x+x^-1)-x(1-x^-2) ]/ (x+x^-1)^2 then disturbted the 2nd x and got (x+x^-1)-(x+x^-1) on the numerator but I think i just found my error; when distributing I turned the x^-2 in the 2nd term to a x^-1 but it should go to a x^-3 right?
 
  • #4
No x^-1 is right.
 
  • #5
is the derivative (2x^-1)/(x+x^-1)^2
 
  • #6
I looked at your solution and I think you solved it right the first time. What made you to doubt it?
 
  • #7
My friend who is working on the same problem got a different answer, and when I graphed the tangent line with that slope it looked like it crossed the graph at 2 points.
 
  • #8
It is okay if it is tangent to the function at the specified point. It doesn't matter how many times it crosses the function at several points. Do you understand the difference between "being tangent" and "crossing"?
 
  • #9
I think i had a sign error my first time around and i made the 2nd x^-1 a negative when it should be positive then you have (x +x^-1 - x + x^-1)/(x+x^-1)^2 shouldn't the +x and -x negate each other but leave you with 2x^-1 for the numerator ?
 
  • #10
ohhh I thought that a tangent line could only hit the graph at one point
 
  • #11
Why would it be +? Isn't (1/x)' = -1/x^2?
 
  • #12
yes it is but then the original derivative is [(x+x^-1) - x(1-x^-2)]/(x+x^-1)^2 so shouldn't the -x turn the x^-2 to +x^-1
 
  • #13
physstudent1 said:
ohhh I thought that a tangent line could only hit the graph at one point
No.

1. A line L can be tangent to a function F at multiple points.
2. A line L can cross a function F at multiple points.
3. Any combination of the two is possible.

Example: y = 1 is tangent to sin(x) at multiple points. y = 1/2 crosses sin(x) at multiple points.
 
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  • #14
oh wow I didn't know that thank you, but shouldn't that -x^-2 term turn to +x^-1
 
  • #15
physstudent1 said:
yes it is but then the original derivative is [(x+x^-1) - x(1-x^-2)]/(x+x^-1)^2 so shouldn't the -x turn the x^-2 to +x^-1
You're right, I missed it. One way to check is to write y = 1/(1 + 1/x^2) then differentiate it. You should get the same answer.
 
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  • #16
heh happens to the best of us thanks for all your help tonight you really cleared up some confused concepts in my thinking
 

FAQ: Derivative and Tangent Line Equation

What is a derivative?

A derivative is a mathematical concept that measures the rate at which one quantity changes in relation to another quantity. In other words, it is the slope of a curve at a specific point.

How is a derivative calculated?

The derivative of a function can be calculated using the limit definition, which involves finding the slope of a secant line as the distance between two points on the curve approaches zero. Alternatively, derivatives can also be calculated using rules and formulas specific to the type of function.

What does the derivative represent?

The derivative represents the instantaneous rate of change of a function at a specific point. This can be interpreted as the slope of the tangent line to the function's curve at that point.

How is a tangent line related to a derivative?

A tangent line is a straight line that touches a curve at only one point. The slope of this line is equal to the derivative of the curve at that point. In other words, the tangent line is the graphical representation of the derivative at a specific point.

What is the importance of the derivative and tangent line?

The derivative and tangent line are fundamental concepts in calculus and are used to solve many real-world problems. They are essential in understanding the behavior of functions and can be used to optimize processes and predict future outcomes.

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