Learn how to find the derivative of ln(x^2). The derivative is d/dx[ln(x^2)].
Below is the graph of f(x) = ln(x^2) 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^2) is a composition of two mathematical structures: the natural logarithm of x squared. This requires the application of both the Logarithmic Rule and the Power Rule. The Logarithmic Rule applies because of the presence of the logarithmic function, and the Power Rule applies because x is raised to the power of 2.
💡 Why this works: The combination of these rules allows us to differentiate ln(x^2) by treating it as a composition of functions and handling each part appropriately.
We first apply the Logarithmic Rule, which tells us that the derivative of ln(u) is 1/u times the derivative of u. Here, u is x^2. Then, we apply the Power Rule to differentiate x^2, yielding 2x. The derivative of ln(x^2) becomes 1/x^2 times 2x.
💡 Why this works: By applying the Logarithmic Rule and the Power Rule, we simplify the expression and obtain the derivative of ln(x^2).
The expression 1/x^2 times 2x simplifies to 2/x. This is the correct derivative of ln(x^2), as verified by ensuring both differentiation rules were properly applied.
💡 Why this works: The simplification process confirms that the derivative of ln(x^2) is 2/x, which is the final form.
[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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