Implicit Diff: Tangent Line at (-3*31/2,1)

In summary, the conversation discusses using implicit differentiation to find an equation of the tangent line to a curve at a given point. The solution involves manipulating the given equation and using the power rule to find the derivative. It is noted that there may be some confusion about plugging in negative numbers, but this should not pose a problem as cube roots of negative numbers are valid. The final answer for the slope is determined to be -3*3^(1/2).
  • #1
Unknown9
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Homework Statement


Implicit Differentiation:Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
x2/3 + y2/3 = 4 at (-3*31/2,1)


Homework Equations


? None?


The Attempt at a Solution


2/3 * x-1/3 + 2/3y-1/3*y' = 0
then after a few steps of switching the sides of the variables

y' = (-2x1/3)/ 2/3*y-1/3

The part that I'm just confused on is putting x into the variables, especially since we're not suppose to use calculators, any help?
 
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  • #2
3*3^(1/2)=3^(3/2). Does that help you simplify the x part?
 
  • #3
Dick said:
3*3^(1/2)=3^(3/2). Does that help you simplify the x part?

I think I got it, does the slope end up being -3*31/2? Otherwise I did it incorrectly :P
 
  • #4
Yes, I think it does. It always helps to show your work when asking question like this.
 
  • #5
Wait wait wait, okay, when I plug in x... how can I plug in a negative number into that...? -3^3/2 into x^- 1/3?
 
  • #6
x^(-1/3) is one over the cube root of x. There isn't any problem with taking the cube roots of negative numbers.
 

FAQ: Implicit Diff: Tangent Line at (-3*31/2,1)

What is implicit differentiation?

Implicit differentiation is a method used in calculus to find the derivative of a function that is not explicitly written in terms of a single variable. It is especially useful when dealing with equations that cannot be easily solved for one variable.

How do you find the tangent line at a specific point using implicit differentiation?

To find the tangent line at a specific point using implicit differentiation, first differentiate the equation with respect to the variable in the equation. Then plug in the given coordinates of the point to find the slope of the tangent line. Finally, use the point-slope formula to write the equation of the tangent line.

What is the significance of (-3*31/2,1) in the context of implicit differentiation?

The point (-3*31/2,1) represents the coordinates of a specific point on the graph of the function. In this case, it is the point at which we are finding the tangent line using implicit differentiation.

Can implicit differentiation be used to find the derivative of any function?

Yes, implicit differentiation can be used to find the derivative of any function, as long as the function is differentiable.

What are some common applications of implicit differentiation?

Implicit differentiation has various applications in fields such as physics, economics, and engineering. It is commonly used to find rates of change, optimization problems, and to analyze curves and surfaces in 3D space.

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