Given the equation:
x^3 + y^3 =
14xy
We need to find the value of the first derivative at the point
(7,7).
First we will use implicit
differentiation.
==> (x^3)' + (y^3)' =
14(xy)'
==> 3x^2 + 3y^2 *y' = 14[
(x'y+xy')]
==> 3x^2 + 3y^2 y' = 14(y +
xy')
==> 3x^2 + 3y^2 y' = 14y +
14xy'
Now we will combine terms with
y'.
==> 3y^2 y' - 14xy' = 14y -
3x^2
==> y' ( 3y^2-14x) = 14y -
3x^2
==> y' = (14y-3x^2)/
(3y^2-14x)
Now we will substitute with
(7,7)
==> y'(7,7) = (14*7 - 3*7^2) / (3*7^2 -
14*7)
= 98-147 / 147-98
= -49/49 =
-1
Then the value of the derivative is
(-1).
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