Find the derivative of $4\sqrt{x} - 2$.

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> [!TLDR]
> The derivative of $4\sqrt{x} - 2$ is found by rewriting $\sqrt{x}$ as $x^{1/2}$, applying the power rule term-by-term, and noting the constant term vanishes. The result is $\frac{2}{\sqrt{x}}$.

The key idea here is that **derivative at a point** is about instantaneous rate of change, but to find a general derivative function we use rules that come from the limit definition. For a function like $4\sqrt{x} - 2$, we don't need to go back to first principles every time — we can use the power rule, which is one of the most reliable tools in differentiation.

The power rule says: if $f(x) = x^n$, then $f'(x) = n x^{n-1}$. This works for any real number $n$, including fractions. That's exactly what we need here, because $\sqrt{x}$ is $x^{1/2}$.

Let's walk through it step by step.

1. **Rewrite the square root as a power.**  
   $\sqrt{x}$ is the same as $x^{1/2}$. So the function becomes:  
   $$f(x) = 4x^{1/2} - 2$$

2. **Differentiate term by term.**  
   The derivative of a sum (or difference) is the sum (or difference) of the derivatives. So we handle $4x^{1/2}$ and $-2$ separately.

3. **Apply the power rule to $4x^{1/2}$.**  
   Bring down the exponent $1/2$ as a coefficient, multiply by the existing constant $4$, then subtract $1$ from the exponent:  
   $$\frac{d}{dx}\left(4x^{1/2}\right) = 4 \cdot \frac{1}{2} x^{(1/2) - 1} = 2x^{-1/2}$$

4. **The derivative of a constant is zero.**  
   The term $-2$ is constant — it doesn't change, so its rate of change is $0$:  
   $$\frac{d}{dx}(-2) = 0$$

5. **Combine the results.**  
   $$f'(x) = 2x^{-1/2} + 0 = 2x^{-1/2}$$ …

Q.Find the derivative of $4\sqrt{x} - 2$ .

Concept understanding — Rules of Differentiation

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Q.Find the derivative of $4\sqrt{x} - 2$ .

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