What is the derivative of ln(x squared)?
Understand the Problem
The question is asking for the derivative of the function ln(x^2), which requires applying the rules of differentiation, particularly the chain rule and the property of logarithms.
Answer
The derivative of $\ln(x^2)$ is $\frac{2}{x}$.
Answer for screen readers
The derivative of the function $\ln(x^2)$ is $\frac{2}{x}$.
Steps to Solve
- Apply the property of logarithms
First, we can simplify the expression using the property of logarithms:
$$ \ln(a^b) = b \cdot \ln(a) $$
Applying this to our function gives:
$$ \ln(x^2) = 2 \cdot \ln(x) $$
- Differentiate the new expression
Now, we can differentiate the simplified function using the derivative of the natural logarithm, which states that if $f(x) = \ln(x)$, then $f'(x) = \frac{1}{x}$.
The derivative of our function becomes:
$$ \frac{d}{dx}[2 \cdot \ln(x)] = 2 \cdot \frac{1}{x} $$
- Write the final derivative
The final expression for the derivative of the original function is:
$$ \frac{d}{dx}[\ln(x^2)] = \frac{2}{x} $$
The derivative of the function $\ln(x^2)$ is $\frac{2}{x}$.
More Information
The property of logarithms allows for simpler differentiation by reducing the power. This technique is often used to find derivatives of logarithmic functions efficiently.
Tips
- Forgetting to apply the property of logarithms before differentiating, leading to a more complicated expression.
- Not remembering that the derivative of $\ln(x)$ is $\frac{1}{x}$, which can lead to incorrect calculations.
