To determine the derivative of the function [tex]\( f(x) = \frac{6}{x} + \sqrt{x} + x^4 \)[/tex] and then evaluate it at [tex]\( x = 1 \)[/tex], follow these steps:
1. Find the derivative of each term individually:
a. For the term [tex]\( \frac{6}{x} \)[/tex]:
[tex]\[
\frac{d}{dx}\left(\frac{6}{x}\right) = \frac{d}{dx}\left(6x^{-1}\right) = 6 \cdot (-1) \cdot x^{-2} = -\frac{6}{x^2}
\][/tex]
b. For the term [tex]\( \sqrt{x} \)[/tex]:
[tex]\[
\frac{d}{dx}\left(\sqrt{x}\right) = \frac{d}{dx}\left(x^{1/2}\right) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}
\][/tex]
c. For the term [tex]\( x^4 \)[/tex]:
[tex]\[
\frac{d}{dx}\left(x^4\right) = 4x^3
\][/tex]
2. Combine these derivatives to find the derivative [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(x) = -\frac{6}{x^2} + \frac{1}{2\sqrt{x}} + 4x^3
\][/tex]
3. Evaluate [tex]\( f'(x) \)[/tex] at [tex]\( x = 1 \)[/tex]:
Substitute [tex]\( x = 1 \)[/tex] into the derivative:
a. For the term [tex]\( -\frac{6}{x^2} \)[/tex]:
[tex]\[
-\frac{6}{1^2} = -6
\][/tex]
b. For the term [tex]\( \frac{1}{2\sqrt{x}} \)[/tex]:
[tex]\[
\frac{1}{2\sqrt{1}} = \frac{1}{2}
\][/tex]
c. For the term [tex]\( 4x^3 \)[/tex]:
[tex]\[
4 \cdot 1^3 = 4
\][/tex]
4. Add these evaluated terms together:
[tex]\[
f'(1) = -6 + \frac{1}{2} + 4
\][/tex]
Convert [tex]\( \frac{1}{2} \)[/tex] to a decimal for easier addition:
[tex]\[
f'(1) = -6 + 0.5 + 4
\][/tex]
[tex]\[
f'(1) = -6 + 4.5
\][/tex]
[tex]\[
f'(1) = -1.5
\][/tex]
Thus, the value of the derivative of the function [tex]\( f(x) \)[/tex] at [tex]\( x = 1 \)[/tex] is [tex]\( \boxed{-1.5} \)[/tex].
