Find the derivative of the algebraic function.
f(x) = 
7x − 3/√x


To find the derivative of the given algebraic function \( f(x) = 7x - \frac{3}{\sqrt{x}} \), let's go step-by-step.

### 1. Rewrite the function
We'll rewrite \( \frac{3}{\sqrt{x}} \) as a power of \( x \).

\[
f(x) = 7x - 3x^{-\frac{1}{2}}
\]

### 2. Differentiate term by term
Now, differentiate each term using basic differentiation rules:
- The derivative of \( 7x \) is \( 7 \).
- The derivative of \( -3x^{-\frac{1}{2}} \) is found using the power rule \( \frac{d}{dx} [x^n] = nx^{n-1} \).

\[
\frac{d}{dx} \left( -3x^{-\frac{1}{2}} \right) = -3 \cdot \left( -\frac{1}{2} \right) x^{-\frac{3}{2}} = \frac{3}{2}x^{-\frac{3}{2}}
\]

### 3. Combine the derivatives
Thus, the derivative of the function is:

\[
f'(x) = 7 + \frac{3}{2}x^{-\frac{3}{2}}
\]

### Final answer:
\[
f'(x) = 7 + \frac{3}{2\sqrt{x^3}}
\]

Would you like more details on any part of this process?

Here are some related questions:
1. What is the power rule for differentiation?
2. How do you differentiate terms with fractional exponents?
3. How do we simplify negative exponents in derivatives?
4. How can derivatives be used to find the slope of a curve?
5. What does the derivative tell us about the behavior of a function?

**Tip:** Always rewrite terms with roots or fractions into exponents when differentiating—it makes the process easier!

Find the derivative of the algebraic function. f(x) = 7x − 3/√x

Learn how to find the derivative of the function f(x) = 7x − 3/√x using the power rule. This step-by-step solution simplifies complex expressions with roots and fractional exponents. Suitable for high school and early college-level students.

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