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Find the first order derivative with respect to
for
.
Evaluate the derivative by following steps:
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 ofwith respect to is .
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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?