What is the derivative of sqrt(3x)?

Steps to Solve

1. Identify the function to differentiate

We are given the function ( f(x) = \sqrt{3x} ). To differentiate this function, we can rewrite it in exponent form.

2. Rewrite the function

Change the square root into an exponent: $$ f(x) = (3x)^{1/2} $$

3. Apply the chain rule

The chain rule states that if you have a composite function ( u(x)^{n} ), the derivative is given by: $$ \frac{d}{dx}[u(x)^{n}] = n \cdot u(x)^{n-1} \cdot u'(x) $$

In our case, ( u(x) = 3x ) and ( n = \frac{1}{2} ).

4. Differentiate the outer function

Using the outer function, we get: $$ f'(x) = \frac{1}{2} (3x)^{-1/2} \cdot \frac{d}{dx}(3x) $$

5. Differentiate the inner function

To find ( \frac{d}{dx}(3x) ), we simply have: $$ \frac{d}{dx}(3x) = 3 $$

6. Combine the results

Now substitute this back into our derivative: $$ f'(x) = \frac{1}{2} (3x)^{-1/2} \cdot 3 $$

7. Simplify the expression

This simplifies to: $$ f'(x) = \frac{3}{2} (3x)^{-1/2} $$

8. Rewrite in terms of square roots

Finally, we can express this as: $$ f'(x) = \frac{3}{2 \sqrt{3x}} $$