Differentiation By Taking Log

To differentiate , we will use logarithmic differentiation.

Step 1: Take the natural logarithm of both sides.

Step 2: Use the logarithmic identity .

Step 3: Differentiate both sides with respect to .
Using the product rule on the right-hand side: On the left-hand side: On the right-hand side, applying the product rule: So, we have:

Step 4: Solve for .
Multiply both sides by : Since , substitute back: Conclusion:
The derivative of is:

We are tasked with finding . The given function is: 1. Step 1: Differentiating To differentiate , we take the natural logarithm of both sides: Now, differentiate implicitly with respect to : Thus: Substitute back: 2. Step 2: Differentiating Let . We differentiate using logarithmic differentiation: Now, differentiate implicitly: Apply the product rule: Thus: 3. Step 3: Final Expression Now, the total derivative of is: Final Answer: