Alright, let's differentiate the given function [tex]\(\frac{d}{dx}\left(\frac{1}{x^2} + 4 \sqrt{x^3}\right)\)[/tex] step-by-step.
1. Identify Each Part of the Function:
We have two terms in the function:
[tex]\[
f(x) = \frac{1}{x^2} + 4 \sqrt{x^3}
\][/tex]
2. Differentiate Each Term Separately:
- For the first term, [tex]\(\frac{1}{x^2}\)[/tex]:
[tex]\[
\frac{1}{x^2} = x^{-2}
\][/tex]
Using the power rule for differentiation, which states that [tex]\(\frac{d}{dx} x^n = nx^{n-1}\)[/tex]:
[tex]\[
\frac{d}{dx} x^{-2} = -2x^{-3} = -\frac{2}{x^3}
\][/tex]
- For the second term, [tex]\(4 \sqrt{x^3}\)[/tex]:
[tex]\[
4 \sqrt{x^3} = 4 (x^3)^{1/2}
\][/tex]
Combine the exponents inside the radical:
[tex]\[
4 (x^3)^{1/2} = 4 x^{3/2}
\][/tex]
Again, applying the power rule:
[tex]\[
\frac{d}{dx} 4 x^{3/2} = 4 \cdot \frac{3}{2} x^{(3/2) - 1} = 4 \cdot \frac{3}{2} x^{1/2}
\][/tex]
Simplify the coefficient:
[tex]\[
4 \cdot \frac{3}{2} = 6
\][/tex]
So,
[tex]\[
\frac{d}{dx} 4 x^{3/2} = 6 x^{1/2} = 6 \sqrt{x}
\][/tex]
3. Combine the Results:
Now, we'll combine the derivatives of each term:
[tex]\[
\frac{d}{dx} \left( \frac{1}{x^2} + 4 \sqrt{x^3} \right) = -\frac{2}{x^3} + 6 \sqrt{x}
\][/tex]
4. Rewrite [tex]\(\sqrt{x}\)[/tex] in terms of [tex]\(x\)[/tex]:
Notice that the expression [tex]\(6 \sqrt{x} (x^{1/2})\)[/tex] can also be represented by [tex]\(6 \frac{\sqrt{x^3}}{x}\)[/tex]:
[tex]\[
\frac{6 \sqrt{x^3}}{x}
\][/tex]
Both forms are equivalent. The final form of the answer is:
[tex]\[
\frac{d}{dx}\left(\frac{1}{x^2} + 4 \sqrt{x^3}\right) = 6 \frac{\sqrt{x^3}}{x} - \frac{2}{x^3}
\][/tex]
Thus, the derivative is:
[tex]\[
\boxed{6 \frac{\sqrt{x^3}}{x} - \frac{2}{x^3}}
\][/tex]
