Implicit differentiation: why apply the Chain Rule?

In summary, the conversation discusses how to obtain the slope at any point on the equation y^2=x without previously clearing y^2. It involves differentiating the equation and treating y as a differentiable function. This can be done by considering y^2 as a composite function and using the Chain Rule. However, the example given is simple enough to just consider x as a function of y and take the inverse to obtain the slope of y as a function of x.
  • #1
mcastillo356
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Homework Statement
Calculate ##dy/dx## if ##y^2=x##
Relevant Equations
Chain Rule
Hi, PF

##y^2=x## is not a function, but it is possible to obtain the slope at any point ##(x,y)## of the equation without previously clearing ##y^2##. It's enough to differentiate respect to ##x## the two members, treat ##y## like a ##x## differentiable function and make use of the Chain Rule to differentiate ##y^2##:

##\dfrac{d}{dx}(y^2)=\dfrac{d}{dx}(x)##

##2y\dfrac{dy}{dx}=1##

##\dfrac{dy}{dx}=\dfrac{1}{2y}##

I can't view ##y^2## like a composite function, instead of just a quadratic expression.

Greetings!
 
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  • #2
You have to consider ##y^2## as the composite of two functions, namely ##t\stackrel{q}{\longmapsto} t^2## after ##x\stackrel{y}{\longmapsto} y(x)## which is ##(q\circ y)\, : \,x \longmapsto (q\circ y)(x)=q(y(x))=y(x)^2.##

Edit: This leads to an equation of the kind ##f(x)=g(x),## namely ##(q\circ y)(x)=\operatorname{id}(x)## which you differentiated on both sides.
 
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  • #3
The examples of implicit differentiation are usually more complicated.
If you are specifically asking about this example, the right side is so simple that it is easy to consider x as a function of y, get the derivative dx/dy = 2y, and take the inverse, dy/dx = 1/(2y) to obtain the slope of y as a function of x.
 
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  • #4
Fine, thanks!
 

FAQ: Implicit differentiation: why apply the Chain Rule?

What is implicit differentiation?

Implicit differentiation is a mathematical technique used to find the derivative of a function that is not explicitly expressed in terms of its independent variable. This is often the case when the function is given in the form of an equation rather than a simple expression.

Why do we need to use the Chain Rule in implicit differentiation?

The Chain Rule is necessary in implicit differentiation because the function being differentiated is not in its simplest form. The Chain Rule allows us to break down the function into smaller parts and find the derivative of each part separately, making it easier to solve for the overall derivative.

How is the Chain Rule applied in implicit differentiation?

The Chain Rule is applied by first identifying the inner function and the outer function of the given equation. The derivative of the outer function is then multiplied by the derivative of the inner function, and this product is then multiplied by the derivative of the inner function with respect to the independent variable. This process is repeated until the entire function has been differentiated.

Can the Chain Rule be used in all cases of implicit differentiation?

Yes, the Chain Rule can be used in all cases of implicit differentiation. It is a fundamental rule of calculus that applies to any composite function, which includes functions in implicit form.

What are some common applications of implicit differentiation?

Implicit differentiation is commonly used in physics and engineering to solve problems involving changing rates, such as velocity and acceleration. It is also used in economics and finance to analyze the relationship between different variables. Additionally, implicit differentiation is used in optimization problems to find the maximum or minimum values of a function.

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