Learn how to find the derivative of ln(x^3). The derivative is d/dx[ln(x^3)].
Below is the graph of f(x) = ln(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 function ln(x^3) requires the use of the Logarithmic Rule because it's a logarithmic function, and the Power Rule because of the exponentiation in x^3. First, we recognize that ln(x^3) involves both a natural logarithm and a power of x.
💡 Why this works: The Logarithmic Rule allows us to differentiate logarithmic functions, while the Power Rule will handle the x^3 term. This combination ensures the correct derivative is calculated.
We start by applying the Logarithmic Rule to ln(x^3), which states that the derivative of ln(u) is 1/u * du/dx. In this case, u = x^3, so we get 1/(x^3) * d/dx[x^3].
💡 Why this works: Next, applying the Power Rule to differentiate x^3, we get 3x^2. This gives us the complete derivative: 1/(x^3) * 3x^2, which simplifies to 3/x.
After applying the rules, we end up with the derivative 3/x. This is the final form of the derivative of ln(x^3), showing how the rate of change behaves with respect to x.
💡 Why this works: By simplifying, we see that the result, 3/x, is correct. We used the proper rules and verified the steps through algebraic manipulation, confirming the derivative is accurate.
[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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