Implicit Derivation: Finding 2nd Derivative of y^2 = x^3

• Bashyboy
In summary, to find the second derivative implicitly of the function y^2 = x^3, we first differentiate both sides to get 2y(dy/dx)=3x^2, and then differentiate again to get 2(dy/dx)^2+2y(d^2y/dx^
Bashyboy

Homework Statement

I am suppose to find the second derivative implicitly of the function y^2 = x^3. I find the first derivative to be dy/dx = 3x^2/2y, but shortly find myself having difficulty in the second derivation. My steps for the second derivative is in the file attached; there are a few additionally steps, but, at the last step, I am in no way nearing the actual answer.

The Attempt at a Solution

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Bashyboy said:

Homework Statement

I am suppose to find the second derivative implicitly of the function y^2 = x^3. I find the first derivative to be dy/dx = 3x^2/2y, but shortly find myself having difficulty in the second derivation. My steps for the second derivative is in the file attached; there are a few additionally steps, but, at the last step, I am in no way nearing the actual answer.

I assume that you result for implicitly differentiating y^2 = x^3, the first time was something like:
$\displaystyle 2y(dy/dx)=3x^2$​
Differentiate that again before solving for dy/dx. It's easier to work with.

Well, I took the derivative from where you advised me to; but I still feel I am getting it wrong. Here are some of the steps I took, though there are few because I had the intuition that I was getting them wrong.

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Bashyboy said:
Well, I took the derivative from where you advised me to; but I still feel I am getting it wrong. Here are some of the steps I took, though there are few because I had the intuition that I was getting them wrong.

That looks OK so far.

Well, the answer is 3x/4y, and I am still not getting this.

SammyS said:

That looks OK so far.

You miss a square in the second step . d/dx(yy')=y'2+yy".

ehild

ehild, I am terribly sorry: I don't really understand what you are saying; I can't not see what it is that I should be fixing in my second step.

ehild, I believe you may have made a mistake when typing with latex.

Do again the derivative of y dy/dx. It is (dy/dx)2+yd2y/dx2.

ehild

SammyS said:

[STRIKE]That looks OK so far[/STRIKE].
I was wrong. As ehild pointed out, the dy/dx should be squared.

$\displaystyle \frac{d}{dx}(2y\frac{dy}{dx})=2\frac{dy}{dx}\frac{dy}{dx}+2y\frac{d^2y}{dx^2}$

What is implicit derivation?

Implicit derivation is a technique used to find the derivative of a function that is not explicitly written in terms of the independent variable. This method is often used when the function is defined implicitly by an equation.

How do you find the second derivative using implicit derivation?

To find the second derivative using implicit derivation, you first take the derivative of both sides of the equation with respect to the independent variable. Then, you solve for the second derivative by isolating the second derivative term on one side of the equation.

Why do we use implicit derivation?

We use implicit derivation when the function is defined implicitly by an equation and cannot be easily solved for the dependent variable. It allows us to find the derivative of a function without having to solve for the dependent variable explicitly.

What are the steps for finding the second derivative using implicit derivation?

The steps for finding the second derivative using implicit derivation are:1. Take the derivative of both sides of the equation with respect to the independent variable.2. Use the chain rule to find the derivative of any functions that are composed within the equation.3. Solve for the second derivative by isolating the second derivative term on one side of the equation.

What are some common examples of implicit derivation?

Some common examples of implicit derivation include finding the second derivative of a circle equation, finding the second derivative of an ellipse equation, and finding the second derivative of a parabola equation.

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