Answer:
To find the equation of the tangent line to the curve \(y = 2x + \frac{5}{x}\) at the point where \(x = 5\), you'll need to follow these steps:
1. Find the slope of the tangent line at \(x = 5\). This slope is equivalent to the derivative of \(y\) with respect to \(x\) evaluated at \(x = 5\).
2. Use the point-slope form of the equation of a line, which is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. In this case, \((x_1, y_1)\) is \((5, 2 \cdot 5 + \frac{5}{5})\).
Let's calculate it step by step:
1. Find the derivative of \(y\) with respect to \(x\):
\[y = 2x + \frac{5}{x}\]
\[y' = 2 - \frac{5}{x^2}\]
2. Evaluate the derivative at \(x = 5\):
\[y'(5) = 2 - \frac{5}{5^2} = 2 - \frac{5}{25} = 2 - \frac{1}{5} = \frac{9}{5}\]
So, the slope of the tangent line at \(x = 5\) is \(\frac{9}{5}\).
3. Now, use the point-slope form of the equation of the tangent line:
\[y - y_1 = m(x - x_1)\]
\[y - \left(2 \cdot 5 + \frac{5}{5}\right) = \frac{9}{5}(x - 5)\]
\[y - (10 + 1) = \frac{9}{5}(x - 5)\]
\[y - 11 = \frac{9}{5}(x - 5)\]
This is the equation of the tangent line to the curve \(y = 2x + \frac{5}{x}\) at the point where \(x = 5\). You can simplify it further if needed.
Explanation: