What is the derivative of sqrt(3x)?
Understand the Problem
The question is asking for the derivative of the function sqrt(3x), which involves applying basic derivative rules from calculus. We will differentiate the function using the chain rule.
Answer
The derivative is \( f'(x) = \frac{3}{2 \sqrt{3x}} \).
Answer for screen readers
The derivative of the function ( \sqrt{3x} ) is ( f'(x) = \frac{3}{2 \sqrt{3x}} ).
Steps to Solve
- 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.
- Rewrite the function
Change the square root into an exponent: $$ f(x) = (3x)^{1/2} $$
- 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} ).
- Differentiate the outer function
Using the outer function, we get: $$ f'(x) = \frac{1}{2} (3x)^{-1/2} \cdot \frac{d}{dx}(3x) $$
- Differentiate the inner function
To find ( \frac{d}{dx}(3x) ), we simply have: $$ \frac{d}{dx}(3x) = 3 $$
- Combine the results
Now substitute this back into our derivative: $$ f'(x) = \frac{1}{2} (3x)^{-1/2} \cdot 3 $$
- Simplify the expression
This simplifies to: $$ f'(x) = \frac{3}{2} (3x)^{-1/2} $$
- Rewrite in terms of square roots
Finally, we can express this as: $$ f'(x) = \frac{3}{2 \sqrt{3x}} $$
The derivative of the function ( \sqrt{3x} ) is ( f'(x) = \frac{3}{2 \sqrt{3x}} ).
More Information
The process of differentiating functions using the chain rule is a fundamental concept in calculus, which helps us understand how rates of change of composite functions are related to their individual components. This function, being a square root, is a common example encountered in calculus.
Tips
- A common mistake is forgetting to apply the chain rule when dealing with nested functions. Remember to differentiate the inner function as well.
- Another mistake is misapplying exponent rules, especially when simplifying after differentiation. Always double-check your algebra.