Learn how to find the derivative of x sqrt(x). The derivative is d/dx[x sqrt(x)].
Below is the graph of f(x) = x sqrt(x) and its derivative f'(x). Notice how the original function and derivative relate to each other.
Original Function:
Derivative:
This derivative represents the rate of change of the original function. The calculation follows standard differentiation rules as shown in the step-by-step solution below.
We begin by analyzing the function f(x) = x sqrt(x). Notice that sqrt(x) can be rewritten as x^(1/2), making the function f(x) = x * x^(1/2) or f(x) = x^(3/2). This allows us to use the Power Rule to differentiate it.
💡 Why this works: The Power Rule is applicable here because we now have a single power function, x^(3/2). Differentiating a power of x follows straightforwardly from the Power Rule, which states that for any term of the form x^n, its derivative is n * x^(n-1).
To differentiate f(x) = x^(3/2), apply the Power Rule directly. The derivative of x^(3/2) is (3/2) * x^(1/2).
💡 Why this works: By applying the Power Rule to x^(3/2), we get the derivative f'(x) = (3/2) * x^(1/2). This is the result of applying the formula for differentiating power functions.
Finally, simplify the expression for the derivative. The result is f'(x) = (3/2) * sqrt(x), which matches the required form.
💡 Why this works: This is the correct derivative for the function f(x) = x sqrt(x), confirming that the differentiation process has been carried out correctly. The Power Rule allows us to efficiently find the rate of change of the function.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ Common Mistake:
Not simplifying the final answer
✅ Correct Approach:
Always simplify your derivative to its most reduced form
Double-check your work by comparing your steps with our solution. Use our calculator to verify each step of the differentiation process.
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