What is the derivative of the square root of 3x?
Understand the Problem
The question is asking for the derivative of the function defined by the square root of 3x. To solve this, we will apply the rules of differentiation specifically focusing on the power rule and the constant multiple 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
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Rewrite the Function in Exponent Form We start with the function ( f(x) = \sqrt{3x} ). We rewrite this using exponent notation: $$ f(x) = (3x)^{1/2} $$
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Apply the Constant Multiple Rule Using the constant multiple rule in differentiation, we take the constant (3) outside of the derivative: $$ f'(x) = \frac{1}{2}(3x)^{-1/2} \cdot (3) $$
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Apply the Chain Rule Now, we apply the chain rule to differentiate ( (3x)^{1/2} ): $$ f'(x) = \frac{1}{2} \cdot 3 \cdot (3x)^{-1/2} \cdot (3) = \frac{3}{2}(3x)^{-1/2} $$
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Simplify the Derivative We simplify the expression: $$ f'(x) = \frac{3}{2\sqrt{3x}} $$
The derivative of the function ( \sqrt{3x} ) is ( f'(x) = \frac{3}{2\sqrt{3x}} ).
More Information
This derivative can be interpreted as the rate of change of the function ( \sqrt{3x} ) with respect to ( x ). Understanding derivatives helps us analyze the behavior of functions, such as finding slopes of tangents at various points.
Tips
- Forgetting to apply the chain rule correctly when differentiating composite functions.
- Not simplifying the final derivative expression fully, leading to a more complex form than necessary.
