Question 840705

1. Differentiate f(x) = {square-root of [(x-1) divided by (x+1)]}.
<pre>
{{{f(x)}}}{{{""=""}}}{{{sqrt((x-1)/(x+1))}}}

{{{matrix(2,1,"",f(x))}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",((x-1)/(x+1))^(1/2))}}}


{{{matrix(2,1,"","f'(x)")}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",
expr(1/2)((x-1)/(x+1))^(-1/2)*derivative_of((x-1)/(x+1)))}}}

{{{matrix(2,1,"","f'(x)")}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",
expr(1/2)((x+1)/(x-1))^(1/2)*(expr(((x+1)(1)-(x-1)(1))/(x+1)^2))))}}}

{{{matrix(2,1,"","f'(x)")}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",
expr(1/2)((x+1)/(x-1))^(1/2)*(expr(((x+1)-(x-1))/(x+1)^2))))}}}

{{{matrix(2,1,"","f'(x)")}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",
expr(1/2)((x+1)/(x-1))^(1/2)*(expr((x+1-x+1)/(x+1)^2))))}}}

{{{matrix(2,1,"","f'(x)")}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",
expr(1/2)((x+1)/(x-1))^(1/2)*(expr(2/(x+1)^2))))}}}

{{{matrix(2,1,"","f'(x)")}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",
expr(1/cross(2))((x+1)/(x-1))^(1/2)*(expr(cross(2)/(x+1)^2))))}}}

{{{matrix(2,1,"","f'(x)")}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",
((x+1)/(x-1))^(1/2)*(expr(1/(x+1)^2))))}}}

{{{"f'(x)"}}}{{{""=""}}}{{{sqrt((x+1)/(x-1))(expr(1/(x+1)^2))}}}
</pre>
2. Differentiate (2y-x)^6 + x^3 = y = 3.
<pre>
I'll assume that last equal sign was a typo and should be a +,
and that you meant
</pre>
2. Differentiate (2y-x)^6 + x^3 = y + 3.
<pre>
{{{(2y-x)^6 + x^3}}}{{{""=""}}}{{{y + 3}}}

{{{6(2y-x)^5(2*expr(dy/dx)-1)+3x^2}}}{{{""=""}}}{{{dy/dx+0}}}

{{{12(2y-x)^5*expr(dy/dx)+18x^2(2y-x)^5}}}{{{""=""}}}{{{dy/dx}}}

Solve for {{{dy/dx}}}

{{{12(2y-x)^5*expr(dy/dx)}}}{{{""-""}}}{{{dy/dx}}} {{{""=""}}} {{{-18x^2(2y-x)^5}}}

{{{(12(2y-x)^5-1)expr(dy/dx)}}}{{{""=""}}}{{{-18x^2(2y-x)^5}}}

{{{dy/dx}}}{{{""=""}}}{{{(-18x^2(2y-x)^5)/(12(2y-x)^5-1)}}}
</pre>
3. Evaluate the derivatives of the function at the point
        3x^3y^2 - 2y3 = -3; (1,2). thank you so much.
<pre>
{{{3x^3y^2 - 2y^3}}}{{{""=""}}}{{{-3}}}

{{{(3x^3*2y*expr(dy/dx)+y^2*x^2)-6y^2*expr(dy/dx)}}}{{{""=""}}}{{{"0"}}}

{{{6x^3y*expr(dy/dx)+y^2*x^2-6y^2*expr(dy/dx)}}}{{{""=""}}}{{{"0"}}}

Solve for {{{dy/dx}}}

{{{6x^3y*expr(dy/dx)-6y^2*expr(dy/dx)}}}{{{""=""}}}{{{-y^2*x^2}}}

{{{(6x^3y-6y^2)*expr(dy/dx)}}}{{{""=""}}}{{{-y^2*x^2}}}

{{{6y(x^3-y)*expr(dy/dx)}}}{{{""=""}}}{{{-y^2*x^2}}}

{{{dy/dx}}}{{{""=""}}}{{{(-y^2*x^2)/(6y(x^3-y))}}}

{{{dy/dx}}}{{{""=""}}}{{{(-y^cross(2)*x^2)/(6cross(y)(x^3-y))}}}

{{{dy/dx}}}{{{""=""}}}{{{(-y*x^2)/(6(x^3-y))}}}

Substitute (x,y) = (1,2).

{{{(dy/dx)["(1,2)"]}}} {{{""=""}}}{{{(-2*1^2)/(6(1^3-2))}}}{{{""=""}}}{{{(-2)/(6(1-2))}}}{{{""=""}}}{{{(-2)/(6*(-1))}}}{{{""=""}}}{{{(-2)/(-6)}}}{{{""=""}}}{{{1/3}}}

Edwin</pre>