To differentiate the function $y = 3x - \sqrt{x}$ using the long method, we will apply the basic rules of differentiation: the power rule and the derivative of the square root function. Let’s go step-by-step.

Step 1: Express the function in terms of exponents.

First, rewrite $\sqrt{x}$ as $x^{1/2}$: $y = 3x - x^{1/2}$

Step 2: Differentiate each term separately.

Now, we differentiate each term using the power rule. The power rule states that if $f(x) = x^n$, then $f'(x) = n \cdot x^{n-1}$.

• For the first term, $3x$: The derivative of $3x$ is simply $3$, since the derivative of $x$ is 1.

• For the second term, $-x^{1/2}$: Use the power rule: the derivative of $x^{1/2}$ is $\frac{1}{2} x^{-1/2}$. Therefore, the derivative of $-x^{1/2}$ is $-\frac{1}{2} x^{-1/2}$.

Step 3: Write the final result.

Now combine the derivatives of the terms: $\frac{dy}{dx} = 3 - \frac{1}{2} x^{-1/2}$

Alternatively, you can rewrite $x^{-1/2}$ as $\frac{1}{\sqrt{x}}$: $\frac{dy}{dx} = 3 - \frac{1}{2\sqrt{x}}$

Final Answer:

$\frac{dy}{dx} = 3 - \frac{1}{2\sqrt{x}}$

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Would you like more details or have any questions about the process?

Related questions:

1. How do you differentiate more complex functions involving exponents or square roots?
2. How does the power rule apply when the exponent is a fraction?
3. Can you use the power rule on functions with negative exponents?
4. How would you differentiate $y = 4x^2 - 5x^{3/2}$?
5. What is the chain rule, and how does it help in differentiation?

Tip: Always simplify the function into powers of $x$ before differentiating—it can make the process easier and clearer.