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Ask by Johnston Floyd. in South Africa
Mar 15,2025

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Find the first order derivative with respect to for .
Evaluate the derivative by following steps:
  • step0: Evaluate the derivative:
  • step1: Use differentiation rules:
  • step2: Find the derivative:
  • step3: Calculate:
  • step4: Substitute back:
  • step5: Calculate:
  • step6: Multiply the terms:

    The derivative of with respect to is .

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Simplify this solution

The Deep Dive

To find when , we can first rewrite the expression using the properties of logarithms. We have:
Now, differentiating with respect to , we get:
So, the derivative is .
Here’s a little fun fact: the natural logarithm has its roots deep in calculus and is connected to exponential growth processes, appearing everywhere from compound interest to population dynamics! Also, when approaches 0 from the right, heads towards infinity—how’s that for an interesting behavior?

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