Implicit Differentiation - Tangent Line & Horizontal Tangents

In summary, the equation of the tangent line to the curve at the point (1, -2) can be found by taking the implicit derivative of y^2 and setting it equal to 0. This results in the equation y+2=3(x-1), or y=3x-5.
  • #1
Pondera
13
0

Homework Statement


2181.jpg


Find an equation of the tangent line to this curve at the point (1, -2).

Homework Equations


The Attempt at a Solution



2y' = 3x^2+6x
y' = 3x^2+6x
y'=3/2x^2+3x

y+2=3(x-1)
y+2=3x-3
y=3x-5
 
Last edited:
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  • #2
Hi Pondera

Pondera said:

Homework Statement


2181.jpg



Homework Equations





The Attempt at a Solution



2y' = 3x^2+6x
y' = 3x^2+6x
where did the 2 go?
Pondera said:
y'=3/2x+3x
where did the power of x go?
Pondera said:
y+2=3(x-1)
y+2=3x-3
y=3x-5

I think the arithmetic needs a little work
 
  • #3
Lane, I divided 3x^2+6x by two and got y'=3/2x^2+3x.

By power of x, I take it you mean power of 2? I neglected to put that in here, but it is on my scratch paper, I appologize.

I don't see where the arithmetic is flawed? I certainly believe that that is likely not the equation/form that I need, but I believe I worked what I figured to be correct out correctly. Can you be more specific?
 
  • #4
ok I don't really understand what you are trying to do, just picked up some misssing parts as discussed

the question is clipped off when I veiw it
 
  • #5
Also the implict derivative w.r.t. x of y2 is not 2y', it will be:

[tex]\frac{d}{dx} y^2 = 2y \frac{dy}{dx} [/tex]

horizontal derivatives where y' = 0
 

FAQ: Implicit Differentiation - Tangent Line & Horizontal Tangents

What is implicit differentiation?

Implicit differentiation is a method used in calculus to find the derivative of an equation that is not written in the form y = f(x). It allows us to find the slope of a tangent line at a given point on a curve.

How do you find the derivative using implicit differentiation?

To find the derivative using implicit differentiation, we treat the dependent variable (usually y) as a function of the independent variable (usually x) and differentiate both sides of the equation with respect to x. This allows us to isolate the derivative and solve for it.

What is the equation for a tangent line?

The equation for a tangent line at a point (x0, y0) on a curve is y - y0 = m(x - x0), where m is the slope of the tangent line. Using implicit differentiation, we can find the slope at a given point and then substitute it into this equation to find the equation of the tangent line.

What is a horizontal tangent?

A horizontal tangent is a line that is parallel to the x-axis and has a slope of 0. This means that the derivative of the function at that point is equal to 0. In other words, the function has no change (or rate of change) at that point.

How do you find horizontal tangents using implicit differentiation?

To find horizontal tangents using implicit differentiation, we set the derivative equal to 0 and solve for the x-values that make the derivative 0. These x-values correspond to the points where the function has a horizontal tangent.

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