Sure, let's find the derivative of the given function step-by-step.
We start with the function:
[tex]\[ y = 4 \sqrt{x} + 5 x^{\frac{1}{5}} \][/tex]
To differentiate this function, we will apply the power rule for derivatives, which states that for any term [tex]\( x^n \)[/tex], the derivative is [tex]\( n x^{n-1} \)[/tex].
First, let's rewrite the square root of [tex]\( x \)[/tex] using exponents:
[tex]\[ \sqrt{x} = x^{\frac{1}{2}} \][/tex]
Now the function becomes:
[tex]\[ y = 4 x^{\frac{1}{2}} + 5 x^{\frac{1}{5}} \][/tex]
Next, we differentiate each term individually.
1. For the first term, [tex]\( 4 x^{\frac{1}{2}} \)[/tex]:
[tex]\[
\frac{d}{dx} \left( 4 x^{\frac{1}{2}} \right) = 4 \cdot \frac{1}{2} x^{\frac{1}{2} - 1} = 2 x^{-\frac{1}{2}}
\][/tex]
2. For the second term, [tex]\( 5 x^{\frac{1}{5}} \)[/tex]:
[tex]\[
\frac{d}{dx} \left( 5 x^{\frac{1}{5}} \right) = 5 \cdot \frac{1}{5} x^{\frac{1}{5} - 1} = x^{-\frac{4}{5}}
\][/tex]
So the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[ \frac{d y}{d x} = 2 x^{-\frac{1}{2}} + x^{-\frac{4}{5}} \][/tex]
Let's rewrite those terms to make them easier to understand. Remembering that:
[tex]\[
x^{-\frac{1}{2}} = \frac{1}{x^{\frac{1}{2}}} = \frac{1}{\sqrt{x}}
\][/tex]
[tex]\[
x^{-\frac{4}{5}} = \frac{1}{x^{\frac{4}{5}}}
\][/tex]
And since [tex]\(\frac{1}{x^{\frac{4}{5}}} = \frac{1}{x^{0.8}} \)[/tex], then:
[tex]\[
\frac{d y}{d x} = \frac{2}{\sqrt{x}} + \frac{1}{x^{0.8}}
\][/tex]
Thus, the final result is:
[tex]\[ \boxed{\frac{d y}{d x} = \frac{1}{x^{0.8}} + \frac{2}{\sqrt{x}}} \][/tex]
