# Implicit Differentiation; Tangent Line

## Homework Statement

Find the equation of a tangent line at the curve at point (-3√3, 1)

x^(1/3) + y^(1/3) = 4

Point-slope:
y-1=m(x-1)

## The Attempt at a Solution

I took the derivative of that equation and resulted in
-y^(2/3)/x^(2/3)

When I tried plugging in x and y to get slope, the equation got very messy and I couldn't get a number out of it. Can anyone help me out?

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Mark44
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## Homework Statement

Find the equation of a tangent line at the curve at point (-3√3, 1)

x^(1/3) = y^(1/3) = 4
Looks like there is a typo here.

## Homework Equations

Point-slope:
y-1=m(x-1)
This would be the line with slope m, that passes through (1, 1).

## The Attempt at a Solution

I took the derivative of that equation and resulted in
y^(2/3)/x^(2/3)
It's hard to tell whether this is correct, since the original equation likely has a typo. If you typed '=' in place of '+', then you have a sign error in your derivative. Also, it would be helpful to have an equation for your derivative; e.g., y' = y^(2/3)/x^(2/3).
When I tried plugging in x and y to get slope, the equation got very messy and I couldn't get a number out of it. Can anyone help me out?

Here's all of my work:

(1/3)x^(-2/3) + 1/3y^(-2/3)(dy/dx) = 0

(dy/dx)1/3y^(2/3) = -1/3x^(2/3)
dy/dx = -3y^(2/3)/3x^(2/3)
dy/dx = -y^(2/3)/x^(2/3)

I hope that helps find my problem. This is where I tried plugging in the x value and everything got complicated.

1/(-3√3)^(2/3)

Mark44
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What is the original equation? I can't tell if your work is correct without knowing the original equation.

Mark44
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(-3√3)^(2/3) can also be written as ((-3√3)^2)^(1/3)

What is the original equation? I can't tell if your work is correct without knowing the original equation.
x^(1/3) + y^(1/3) = 4

This is the original equation. Sorry about the typo before.

Mark44
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That's what I thought, but I wanted to make sure. You should have enough information to complete the problem now.

(-3√3)^(2/3) can also be written as ((-3√3)^2)^(1/3)
Just to make sure, the slope is 1/3?

Mark44
Mentor
I get -1/3.