https://file.aiquickdraw.com/mathbot/file/1726890185392l8o4mhaf.jpg

Let's go through the steps to solve the problem.

### Part a: Find \( f'(a) \)
The function is given as:

\[
f(x) = \frac{4}{\sqrt{x}} = 4x^{-\frac{1}{2}}
\]

To find \( f'(x) \), we apply the power rule of differentiation:

\[
f'(x) = \frac{d}{dx} \left( 4x^{-\frac{1}{2}} \right) = 4 \cdot \left( -\frac{1}{2} \right) x^{-\frac{3}{2}} = -2x^{-\frac{3}{2}}
\]

Now, substitute \( a = \frac{1}{16} \) into \( f'(x) \):

\[
f'\left( \frac{1}{16} \right) = -2 \left( \frac{1}{16} \right)^{-\frac{3}{2}} 
\]

We'll calculate this value next.

### Part b: Tangent Line Equation
The equation of a tangent line at the point \( (a, f(a)) \) is given by:

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

We already have \( a = \frac{1}{16} \). Now, we need to calculate \( f(a) \) and use the derivative \( f'(a) \) from part a to find the equation of the tangent line. Let's calculate these values next.### Part a: Finding \( f'(a) \)
We found that:

\[
f'\left( \frac{1}{16} \right) = -128
\]

### Part b: Tangent Line Equation
To determine the tangent line equation at the point \( \left(\frac{1}{16}, f\left(\frac{1}{16}\right)\right) \), we use the point \( \left( \frac{1}{16}, 16 \right) \) and the slope \( f'(a) = -128 \).

The tangent line equation is:

\[
y - 16 = -128 \left( x - \frac{1}{16} \right)
\]

Simplifying:

\[
y - 16 = -128x + 8
\]

Thus, the equation of the tangent line is:

\[
y = -128x + 24
\]

Let me know if you would like further clarifications or details.

### Related Questions:
1. How is the power rule for differentiation applied to different exponents?
2. What is the geometric significance of a tangent line to a curve?
3. How do you find the equation of a tangent line at a specific point?
4. Can we apply the chain rule to differentiate more complex expressions like \( f(x) = \frac{1}{x^n} \)?
5. What is the derivative of functions involving higher roots, such as cube or fourth roots?

### Tip:
When calculating derivatives involving fractional powers, make sure to simplify the expression into a form that can be easily differentiated using basic rules like the power rule.

For the function f(x) = 4/√x and a = 1/16, find f'(a) and determine the equation of the tangent line at the point (a, f(a)).

This solution explains how to find the derivative of f(x) = 4/√x using the power rule and how to determine the equation of the tangent line at the point a = 1/16. Suitable for students in Grades 11-12, this problem covers key calculus concepts including derivatives and the tangent line equation.

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