To find the equation of the tangent line at a given point on a curve, we need to follow these steps:
- Differentiate the given equation implicitly with respect to \(x\) to find \(\frac{dy}{dx}\).
- Substitute the given value of \(x\) into the original equation to find the corresponding \(y\) value.
- Substitute the \(x\) and \(y\) values into the derivative to find the slope of the tangent line.
- Use the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point on the curve, to find the equation of the tangent line.
Given the equation of the curve:
\[
2y^3(x-5) + x\sqrt{y} = 10
\]
Substitute \( x = 5 \) into the equation:
\[
2y^3(5-5) + 5\sqrt{y} = 10
\]
This simplifies to:
\[
5\sqrt{y} = 10
\]
Divide both sides by 5:
\[
\sqrt{y} = 2
\]
Square both sides to solve for \( y \):
\[
y = 4
\]
Differentiate the equation \( 2y^3(x-5) + x\sqrt{y} = 10 \) with respect to \( x \):
Differentiate \( 2y^3(x-5) \):
\[
\frac{d}{dx}[2y^3(x-5)] = 2y^3 \cdot 1 + (x-5) \cdot 6y^2 \frac{dy}{dx} = 2y^3 + 6y^2(x-5)\frac{dy}{dx}
\]
Differentiate \( x\sqrt{y} \):
\[
\frac{d}{dx}[x\sqrt{y}] = \sqrt{y} + x \cdot \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = \sqrt{y} + \frac{x}{2\sqrt{y}}\frac{dy}{dx}
\]
Combine these results:
\[
2y^3 + 6y^2(x-5)\frac{dy}{dx} + \sqrt{y} + \frac{x}{2\sqrt{y}}\frac{dy}{dx} = 0
\]
Substitute \( x = 5 \) and \( y = 4 \) into the differentiated equation:
\[
2(4)^3 + 6(4)^2(5-5)\frac{dy}{dx} + \sqrt{4} + \frac{5}{2\sqrt{4}}\frac{dy}{dx} = 0
\]
Simplify:
\[
128 + 2 + \frac{5}{4}\frac{dy}{dx} = 0
\]
\[
130 + \frac{5}{4}\frac{dy}{dx} = 0
\]
Solve for \(\frac{dy}{dx}\):
\[
\frac{5}{4}\frac{dy}{dx} = -130
\]
\[
\frac{dy}{dx} = -130 \times \frac{4}{5}
\]
\[
\frac{dy}{dx} = -104
\]
The equation of the tangent line at a point \((x_0, y_0)\) is given by:
\[
y - y_0 = m(x - x_0)
\]
where \( m = \frac{dy}{dx} \), \( x_0 = 5 \), and \( y_0 = 4 \).
Substitute the values:
\[
y - 4 = -104(x - 5)
\]
Simplify:
\[
y - 4 = -104x + 520
\]
\[
y = -104x + 524
\]
The equation of the tangent line is:
\[
\boxed{y = -104x + 524}
\]
Answered
