help me with finding equation of tangent (implicit differentiation)

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October 17th 2009, 05:30 PM
zpwnchen

help me with finding equation of tangent (implicit differentiation)

please someone help me with this:

Quote:

Consider the curve http://webwork.asu.edu/webwork2_file...94917d6d91.png
The equation of the tangent line to the curve at the point http://webwork.asu.edu/webwork2_file...d47ccf5351.png has the form http://webwork.asu.edu/webwork2_file...0ea661f941.png where
http://webwork.asu.edu/webwork2_file...326d834761.png_________and http://webwork.asu.edu/webwork2_file...7973f0e8c1.png_________________
October 17th 2009, 06:16 PM
redsoxfan325

Quote:

Originally Posted by zpwnchen

please someone help me with this:

Consider the curve http://webwork.asu.edu/webwork2_file...94917d6d91.png
The equation of the tangent line to the curve at the point http://webwork.asu.edu/webwork2_file...d47ccf5351.png has the form http://webwork.asu.edu/webwork2_file...0ea661f941.png where
http://webwork.asu.edu/webwork2_file...326d834761.png_________and http://webwork.asu.edu/webwork2_file...7973f0e8c1.png_________________

Differentiate implicitly:

$6x^5+2x\frac{dy}{dx}+2y+4y^3\frac{dy}{dx}=0 \implies \frac{dy}{dx}=-\frac{6x^5+2y}{2x+4y^3}$

Plug in $(1,1)$ to find the slope and then use the point-slope equation to find the tangent line.
October 17th 2009, 06:25 PM
zpwnchen

Why do i have to use implicit differentiation?
$<br /> m = f_{x} (x,y)$ so $F_x = 6x^5+2y = 6+2 =8$ .. why is not right?
October 17th 2009, 06:31 PM
mr fantastic

Quote:

Originally Posted by zpwnchen

Why do i have to use implicit differentiation?
$<br /> m = f_(x) (x,y)$ so $F_x = 6x^5+2y = 6+2 =8$ .. why is not right?

.
This is wrong. You require the value of dy/dx at the given point. Note that y is an implicit function of x.
October 17th 2009, 06:32 PM
redsoxfan325

Quote:

Originally Posted by zpwnchen

Why do i have to use implicit differentiation?
$<br /> m = f_(x) (x,y)$ so $F_x = 6x^5+2y = 6+2 =8$ .. why is not right?

This is not a function of three variables (i.e. $z=f(x,y)$ ) and therefore partial derivatives are not necessary because you are only trying to find $\frac{dy}{dx}$ .
October 17th 2009, 06:39 PM
zpwnchen

Quote:

Originally Posted by zpwnchen

Why do i have to use implicit differentiation?
$<br /> m = f_{x} (x,y)$ so $F_x = 6x^5+2y = 6+2 =8$ .. why is not right?

So this is right when y is not included in the function?
October 17th 2009, 06:41 PM
zpwnchen

Please delete it
October 17th 2009, 06:46 PM
mr fantastic

Quote:

Originally Posted by zpwnchen

So this is right when y is not included in the function?

Right for what? This makes no sense.

You are clearly confusing functions of several variables with functions of a single variable. To get the tangent to a curve defined by the relation f(x, y) = c you need to calculate dy/dx. To do this you either make y the subject (where possble or convenient) and differentiate or you just differentiate the relation using implicit differentiation and then solve for dy/dx. Either way, f(x, y) = c defines a function of a single variable.
October 17th 2009, 06:56 PM
zpwnchen

i mean. if it's f(x) function(single variable) then i do not need to implicitly differentiate and just $F_{x}$ would give me m right?
Because it is a f(x y) function (several variable) so i have to implicit differentiate dy/dx to get m right?
October 17th 2009, 07:05 PM
redsoxfan325

Quote:

Originally Posted by zpwnchen

i mean. if it's f(x) function(single variable) then i do not need to implicitly differentiate and just $F_{x}$ would give me m right?
Because it is a f(x y) function (several variable) so i have to implicit differentiate dy/dx to get m right?

You could think of the function as $x^6+2xf(x)+f^4(x)=4$ .

That, you can see, is a function of one variable. If you could solve for $f(x)$ , than you could just take a regular derivative, but since solving for $f(x)$ is either very hard or impossible, it makes the most sense to differentiate implicitly.