Find the derivative for the given function:

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Mar 21,2025

Question

Find the derivative for the given function: $$ f(x)=\frac{1}{\sqrt{x}}-\frac{10}{x^{2}}+8 $$ $$ f^{\prime}(x)= $$

UpStudy AI · Accepted Answer

We start with the function
$$
f(x)=\frac{1}{\sqrt{x}}-\frac{10}{x^{2}}+8.
$$

**Step 1. Rewrite the function using exponents.**

Recall that
$$
\frac{1}{\sqrt{x}}=x^{-\frac{1}{2}} \quad \text{and} \quad \frac{10}{x^2}=10x^{-2}.
$$
Thus, the function becomes
$$
f(x)=x^{-\frac{1}{2}}-10x^{-2}+8.
$$

**Step 2. Differentiate term by term using the power rule.**

The power rule states that if $$ g(x)=x^n $$, then
$$
g'(x)=nx^{n-1}.
$$

1. For the first term $$ x^{-\frac{1}{2}} $$:
   $$
   \frac{d}{dx}\left(x^{-\frac{1}{2}}\right)=-\frac{1}{2}x^{-\frac{3}{2}}.
   $$

2. For the second term $$-10x^{-2}$$:
   $$
   \frac{d}{dx}\left(-10x^{-2}\right)=-10\cdot(-2)x^{-3}=20x^{-3}.
   $$

3. For the constant $$8$$:
   $$
   \frac{d}{dx}(8)=0.
   $$

**Step 3. Combine the derivatives.**

Adding the results together gives
$$
f'(x)=-\frac{1}{2}x^{-\frac{3}{2}}+20x^{-3}.
$$

**Final Answer:**
$$
f^{\prime}(x)=-\frac{1}{2}x^{-\frac{3}{2}}+20x^{-3}.
$$
$$ f^{\prime}(x) = -\frac{1}{2}x^{-\frac{3}{2}} + 20x^{-3} $$

Find the derivative for the given function:

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