Chapter 3 | Derivatives
315
Solution Begin by finding
dy dx .
⎛ ⎝ y 3 + x 3 −3 xy ⎞
d dx
⎠ = d ⎠ = 0
dx (0)
⎛ ⎝ 3 y +
x ⎞
dy dx +3
dy dx 3
3 y 2
x 2 −
y −3 x 2 3 y 2 −3 x .
dy dx = 3
⎛ ⎝ 3
⎞ ⎠ into
y −3 x 2 3 y 2 −3 x
dy dx = 3
3 2
Next, substitute
to find the slope of the tangent line:
2 ,
dy dx | ⎛
=−1.
⎞ ⎠
⎝ 3 2
, 3 2
Finally, substitute into the point-slope equation of the line to obtain y =− x +3.
Example 3.73 Applying Implicit Differentiation
In a simple video game, a rocket travels in an elliptical orbit whose path is described by the equation 4 x 2 +25 y 2 =100. The rocket can fire missiles along lines tangent to its path. The object of the game is to destroy an incoming asteroid traveling along the positive x -axis toward (0, 0). If the rocket fires a missile when it is located at ⎛ ⎝ 3, 8 5 ⎞ ⎠ , where will it intersect the x -axis?
Solution To solve this problem, we must determine where the line tangent to the graph of 4 x 2 +25 y 2 =100 at ⎛ ⎝ 3, 8 5 ⎞ ⎠ intersects the x -axis. Begin by finding dy dx
implicitly.
Differentiating, we have
dy dx =0.
8 x +50 y
dy dx ,
Solving for
we have
dy dx = − 4
x 25 y
.
dy dx |
⎛ ⎝ 3, 8 5
⎞ ⎠
= − 3 10 .
The equation of the tangent line is y = − 3 10
x + 5
The slope of the tangent line is
To
2 .
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