Simple Power Rule in Implicit Differentiation

Hi all,

I have just come across implicit differentiation in my study of calculus and I am slightly confused. Why do I have to use the chain rule to differentiate a function implicitly? More importantly, why CAN'T I use the simple power rule?

Thanks!

Re: Simple Power Rule in Implicit Differentiation

In most cases we differentiate with respect to a certain variable. But if the function is not explicitly in terms of the variable in question you need to use the chain rule.

These tutes are really good for your understanding

Implicit Differentiation

Visual Calculus - Implicit Differentiation

Re: Simple Power Rule in Implicit Differentiation

Yes, I understand that. That is the rule stated in my textbook, but I don't understand why it is true. It seems arbitrary. WHY do we use the chain rule when "the function is not explicitly in terms of the variable in question," rather than the simple power rule?

Re: Simple Power Rule in Implicit Differentiation

A function not explicitly (but implicitly) in terms of x is a function of a function of x, i.e. a composite function of x.

And when do we ever use the chain rule?

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Btw... don't integrate - balloontegrate!

... though, in this case, I should say, don't differeniate - balloontiate! E.g., the chain rule...

http://www.ballooncalculus.org/asy/chain.png

Balloon Calculus; standard integrals, derivatives and methods

Re: Simple Power Rule in Implicit Differentiation

I understand that a function can be defined implicitly in terms of x, but can someone please explain to me why I can't use the simple power rule to implicitly differentiate? Please can someone explain this to me; this is so frustrating. Maybe I should clear up my question a little bit: why is: d/dx [yˆ2] = 2y* dy/dx and not just 2y.

Re: Simple Power Rule in Implicit Differentiation

bottom line is that certain variables can represent entire functions.

consider the following example ...

$\displaystyle \frac{d}{dx}\left[(x^2+1)^3] = 3(x^2+1)^2 \cdot 2x$

$\displaystyle \frac{d}{dx}\left[y^3 \right] = 3y^2 \cdot \frac{dy}{dx}$

... can you see these two derivatives are one and the same if $\displaystyle y = x^2+1$ ?

Re: Simple Power Rule in Implicit Differentiation

Thank you skeeter. That was definitely the most helpful reply, but I just have one more question. Is there anything in the *definition* of the simple power rule that suggests you can't use it when implicitly differentiating a function?

Re: Simple Power Rule in Implicit Differentiation

Quote:

Originally Posted by

**nicksbyman** Thank you skeeter. That was definitely the most helpful reply, but I just have one more question. Is there anything in the *definition* of the simple power rule that suggests you can't use it when implicitly differentiating a function?

note that the "simple power rule" is used in the example I gave ... it is not mutually exclusive with the chain rule. The chain rule is applicable for all derivative rules.