Learn how to find the derivative of x^3. The derivative is 3x².
Below is the graph of f(x) = x^3 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.
The Power Rule applies to functions of the form x^n, where n is a constant. Since x^3 fits this form, we can directly apply the Power Rule to differentiate it.
💡 Why this works: The function x^3 has the structure of a power function with n = 3. The Power Rule allows us to differentiate such functions by multiplying the exponent by the coefficient and reducing the exponent by 1.
To differentiate x^3 using the Power Rule, multiply the exponent (3) by the coefficient (which is 1 for x^3) and then subtract 1 from the exponent.
💡 Why this works: This results in f'(x) = 3x², as we take the exponent 3, multiply it by the coefficient 1, and decrease the exponent by 1 to get 2.
The simplified derivative of x^3 is 3x², which is the final result. This form is the correct derivative, reflecting the slope of the tangent line to the curve at any point.
💡 Why this works: By applying the Power Rule and simplifying, we confirm that the derivative of x^3 is 3x². This shows the rate at which x^3 changes with respect to x.
[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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