Of course! Let's differentiate the given function step-by-step:
Given function:
[tex]\[ y = \sqrt[3]{x} + \frac{1}{\sqrt{x}} - 3 \][/tex]
### Step 1: Rewrite the function using exponents
[tex]\[ \sqrt[3]{x} = x^{1/3} \][/tex]
[tex]\[ \frac{1}{\sqrt{x}} = x^{-1/2} \][/tex]
Thus, the given function can be rewritten as:
[tex]\[ y = x^{1/3} + x^{-1/2} - 3 \][/tex]
### Step 2: Differentiate each term individually
To differentiate [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex], we will use the power rule:
[tex]\[ \frac{d}{dx} x^n = n x^{n-1} \][/tex]
#### Differentiate [tex]\( x^{1/3} \)[/tex]:
[tex]\[ \frac{d}{dx} x^{1/3} = \frac{1}{3} x^{1/3 - 1} = \frac{1}{3} x^{-2/3} \][/tex]
#### Differentiate [tex]\( x^{-1/2} \)[/tex]:
[tex]\[ \frac{d}{dx} x^{-1/2} = -\frac{1}{2} x^{-1/2 - 1} = -\frac{1}{2} x^{-3/2} \][/tex]
#### Differentiate [tex]\(-3\)[/tex]:
The derivative of a constant is 0, so:
[tex]\[ \frac{d}{dx} (-3) = 0 \][/tex]
### Step 3: Combine the derivatives
Putting it all together, the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[ \frac{dy}{dx} = \frac{1}{3} x^{-2/3} - \frac{1}{2} x^{-3/2} \][/tex]
### Step 4: Simplify the expression
Expressing the derivative terms with positive exponents in the denominator:
[tex]\[ \frac{1}{3} x^{-2/3} = \frac{1}{3 x^{2/3}} \][/tex]
[tex]\[ -\frac{1}{2} x^{-3/2} = -\frac{1}{2 x^{3/2}} \][/tex]
Thus, the simplified derivative is:
[tex]\[ \frac{dy}{dx} = \frac{1}{3 x^{2/3}} - \frac{1}{2 x^{3/2}} \][/tex]
### Numerical Form:
Finally, substituting the approximations of fractional powers:
[tex]\[ \frac{dy}{dx} \approx 0.333333333333333/x^{0.666666666666667} - 0.5/x^{1.5} \][/tex]
So, the final detailed step-by-step differentiation result is:
[tex]\[ \frac{dy}{dx} = \frac{1}{3} x^{-2/3} - \frac{1}{2} x^{-3/2} \approx \frac{0.333333333333333}{x^{0.666666666666667}} - \frac{0.5}{x^{1.5}} \][/tex]
This corresponds to the result: [tex]\(-0.5/x^{1.5} + 0.333333333333333/x^{0.666666666666667}\)[/tex].
