To solve for [tex]\( y' \)[/tex] and find the slope of the tangent line at the point [tex]\((2, 2)\)[/tex] for the given equation
[tex]\[ \sqrt{12 + y^2} - x^3 + 4 = 0, \][/tex]
follow these steps:
### Step 1: Implicit Differentiation
Differentiate both sides of the equation with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{d}{dx} \left( \sqrt{12 + y^2} - x^3 + 4 \right) = \frac{d}{dx}(0). \][/tex]
The left-hand side simplifies using the chain rule.
1. For the term [tex]\( \sqrt{12 + y^2} \)[/tex]:
[tex]\[ \frac{d}{dx} \left( \sqrt{12 + y^2} \right) = \frac{1}{2\sqrt{12 + y^2}} \cdot \frac{d}{dx}(12 + y^2) = \frac{1}{2\sqrt{12 + y^2}} \cdot 2y \cdot \frac{dy}{dx}. \][/tex]
Therefore,
[tex]\[ \frac{d}{dx} \left( \sqrt{12 + y^2} \right) = \frac{y}{\sqrt{12 + y^2}} \cdot y'. \][/tex]
2. For the term [tex]\( -x^3 \)[/tex]:
[tex]\[ \frac{d}{dx} \left( -x^3 \right) = -3x^2. \][/tex]
3. The term [tex]\( +4 \)[/tex] is a constant, so its derivative is [tex]\( 0 \)[/tex].
Putting it all together, we have:
[tex]\[ \frac{y}{\sqrt{12 + y^2}} \cdot y' - 3x^2 = 0. \][/tex]
### Step 2: Solve for [tex]\( y' \)[/tex]
To isolate [tex]\( y' \)[/tex]:
[tex]\[ \frac{y}{\sqrt{12 + y^2}} \cdot y' = 3x^2. \][/tex]
[tex]\[ y' = \frac{3x^2 \sqrt{12 + y^2}}{y}. \][/tex]
### Step 3: Find [tex]\( y' \)[/tex] at the Point [tex]\( (2, 2) \)[/tex]
Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 2 \)[/tex] into the equation for [tex]\( y' \)[/tex]:
[tex]\[ y' = \frac{3(2)^2 \sqrt{12 + (2)^2}}{2}. \][/tex]
Calculate each part:
[tex]\[ (2)^2 = 4, \][/tex]
[tex]\[ 12 + (2)^2 = 12 + 4 = 16, \][/tex]
[tex]\[ \sqrt{16} = 4. \][/tex]
Now substitute back:
[tex]\[ y' = \frac{3 \cdot 4 \cdot 4}{2} = \frac{48}{2} = 24. \][/tex]
### Summary
Therefore, the derivative [tex]\( y' \)[/tex] in general form is:
[tex]\[ y' = \frac{3x^2 \sqrt{12 + y^2}}{y}, \][/tex]
and at the point [tex]\( (2, 2) \)[/tex], the value of [tex]\( y' \)[/tex] (the slope of the tangent line) is:
[tex]\[ \left.y^{\prime}\right|_{(2, 2)} = 24. \][/tex]
So,
[tex]\[ \begin{array}{l}
y^{\prime}=\frac{3x^2 \sqrt{12 + y^2}}{y} \\
\left.y^{\prime}\right|_{(2,2)}=24
\end{array} \][/tex]
