Implicit Differentiation

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March 25th 2008, 07:12 PM
lyzzi3

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Implicit Differentiation

I think my teacher was talking about turning y into y(x) or something but I'm just completely lost on this one
March 25th 2008, 07:23 PM
TheEmptySet

$5x^2+4x+xy=2$ using implicit we get

$10x+4+y+xy'=0$

$y(2)=-13$

Solving for y' we get

$y'=- \left( \frac{10x+4+y}{x} \right)$

evaluate at x=2

$y'(2)=-\left( \frac{10(2)+4+(-13)}{2} \right)=-\frac{11}{2}=5.5$
March 25th 2008, 07:26 PM
o_O

Let's do an example to illustrate implicit differentiation:

$x^{2} + y^{3} = 9$

Now, if you were to ask to find the derivative of this, you could solve for y then differentiate the equation with respect to x. However, not all functions are easily represented only in terms of x (ex. you could not solve for y for this equation: $x^{2} + xy + y^{2} = 5$ ). Fortunately, we are still able to find the derivative to such functions.

Note that y is a function of x so that we can differentiate it normally. However, we would have to use the chain rule. I think if you study the example, you'll get what I mean:

$x^{2} + y^{3} = 9$
$2x + \underbrace{3y^{2}y'}_{} = 0 \quad \mbox{Differentiated with respect to x}$

Looking at the underbrace, notice how we used the power rule (3y^2) and in addition the chain rule (hence the y'). Continuing on to solve for y':
$3y^{2}y' = -2x$
$y' = \frac{-2x}{3y^{2}}$

Note it's perfectly fine to have both x and y in our solution to y'.

That is basically the trick when implicitly differentiating equations. See if you can extend this to your question.
March 25th 2008, 07:33 PM
SengNee

Quote:

Originally Posted by lyzzi3

I think my teacher was talking about turning y into y(x) or something but I'm just completely lost on this one

$5x^2+4x+xy(x)=2$
$10x+4+y(x)+x[y'(x)]=0$

If, $y(2)=-13$

$10(2)+4+(-13)+(2)[y'(2)]=0$
$11+2[y'(2)]=0$
$y'(2)=-\frac{11}{2}$