Find an equation of the tangent line to the curve f(x) = x root x + cos ^2 x + 2 e^-x

To find the equation of the tangent line to the curve \( f(x) = x \sqrt{x} + \cos^2(x) + 2e^{-x} \) at a given point \( x = a \), we follow these steps:

1. **Find \( f(a) \)**: Evaluate the function at \( x = a \).
2. **Find \( f'(x) \)**: Compute the derivative of the function.
3. **Find \( f'(a) \)**: Evaluate the derivative at \( x = a \).
4. **Equation of the tangent line**: Use the point-slope form of the line equation: \( y - f(a) = f'(a)(x - a) \).

### Step 1: Find \( f(a) \)
Let \( a \) be the point at which we want the tangent line. We need to find \( f(a) \):

\[ f(a) = a \sqrt{a} + \cos^2(a) + 2e^{-a} \]

### Step 2: Find \( f'(x) \)
Compute the derivative \( f'(x) \).

\[ f(x) = x \sqrt{x} + \cos^2(x) + 2e^{-x} \]

1. Derivative of \( x \sqrt{x} \):
   \[ x \sqrt{x} = x \cdot x^{1/2} = x^{3/2} \]
   \[ \frac{d}{dx}(x^{3/2}) = \frac{3}{2}x^{1/2} \]

2. Derivative of \( \cos^2(x) \) using the chain rule:
   \[ \frac{d}{dx}(\cos^2(x)) = 2\cos(x) \cdot (-\sin(x)) = -2\cos(x)\sin(x) \]

3. Derivative of \( 2e^{-x} \):
   \[ \frac{d}{dx}(2e^{-x}) = 2 \cdot (-e^{-x}) = -2e^{-x} \]

Combine these results:

\[ f'(x) = \frac{3}{2}x^{1/2} - 2\cos(x)\sin(x) - 2e^{-x} \]

### Step 3: Find \( f'(a) \)
Evaluate the derivative at \( x = a \):

\[ f'(a) = \frac{3}{2}a^{1/2} - 2\cos(a)\sin(a) - 2e^{-a} \]

### Step 4: Equation of the tangent line
The equation of the tangent line at \( x = a \) is given by:

\[ y - f(a) = f'(a)(x - a) \]

Substituting \( f(a) \) and \( f'(a) \):

\[ y - \left( a \sqrt{a} + \cos^2(a) + 2e^{-a} \right) = \left( \frac{3}{2}a^{1/2} - 2\cos(a)\sin(a) - 2e^{-a} \right)(x - a) \]

This is the equation of the tangent line to the curve at \( x = a \).

Do you want any more details or have any specific questions about this problem?

Here are five more questions you might be interested in:
1. How do you find the second derivative of the given function?
2. Can you find the critical points of the function \( f(x) \)?
3. How do you determine the concavity of the function \( f(x) \)?
4. What are the inflection points of the function \( f(x) \)?
5. How do you find the local maxima and minima of \( f(x) \)?

**Tip:** Always check the domain of the function before finding derivatives and solving equations to ensure the calculations are valid.

Learn how to find the equation of the tangent line to the curve f(x) = x sqrt(x) + cos^2(x) + 2e^-x at a given point using calculus. This problem involves computing derivatives and applying the point-slope form of a line equation. Suitable for college-level students and those studying calculus.

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