To find the derivative [tex]\( f'(a) \)[/tex] at the given number [tex]\( a \)[/tex], let's follow the detailed steps:
### Step 1: Define the function
The function given is:
[tex]\[ f(x) = \sqrt{3x + 1} \][/tex]
### Step 2: Find the derivative [tex]\( f'(x) \)[/tex]
To find the derivative, we use the chain rule. The chain rule states that if you have a composite function [tex]\( f(g(x)) \)[/tex], then the derivative [tex]\( f'(x) \)[/tex] is [tex]\( f'(g(x)) \cdot g'(x) \)[/tex].
Let [tex]\( u = 3x + 1 \)[/tex]. Then, [tex]\( f(x) = \sqrt{u} \)[/tex].
First, find the derivative of [tex]\( \sqrt{u} \)[/tex] with respect to [tex]\( u \)[/tex]:
[tex]\[ \frac{d}{du} (\sqrt{u}) = \frac{1}{2\sqrt{u}} \][/tex]
Next, find the derivative of [tex]\( u = 3x + 1 \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{du}{dx} = 3 \][/tex]
Now, apply the chain rule:
[tex]\[ f'(x) = \frac{d}{dx} (\sqrt{3x + 1}) = \frac{d}{du} (\sqrt{u}) \cdot \frac{du}{dx} = \frac{1}{2\sqrt{3x + 1}} \cdot 3 = \frac{3}{2\sqrt{3x + 1}} \][/tex]
So, the derivative of [tex]\( f(x) \)[/tex] is:
[tex]\[ f'(x) = \frac{3}{2\sqrt{3x + 1}} \][/tex]
### Step 3: Evaluate the derivative at [tex]\( x = a \)[/tex]
Given [tex]\( a = 5 \)[/tex], we need to evaluate [tex]\( f'(5) \)[/tex].
Substitute [tex]\( x = 5 \)[/tex] into the derivative:
[tex]\[ f'(5) = \frac{3}{2\sqrt{3(5) + 1}} = \frac{3}{2\sqrt{15 + 1}} = \frac{3}{2\sqrt{16}} = \frac{3}{2 \cdot 4} = \frac{3}{8} \][/tex]
### Conclusion
The derivative of the function [tex]\( f(x) = \sqrt{3x + 1} \)[/tex] at [tex]\( a = 5 \)[/tex] is:
[tex]\[ f'(5) = \frac{3}{8} \][/tex]
This gives us the exact value of the derivative at the specified point.
