Pregunta
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The first partial derivative with respect to
is
, and the first partial derivative with respect to
is
.
Solución
We are given
Step 1: Find the partial derivative with respect to
.
Recall that when differentiating an exponential function, we apply the chain rule. For a function of the form
, the partial derivative with respect to
is
Here, the inner function is
Differentiate
with respect to
:
Thus, the partial derivative of
with respect to
is
Step 2: Find the partial derivative with respect to
.
Similarly, apply the chain rule for the derivative with respect to
:
Differentiate
with respect to
:
Thus, the partial derivative of
with respect to
is
Final Answers:
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Como
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The Deep Dive
To find the first partial derivatives of the function
, we can apply the chain rule.
First, for the partial derivative with respect to
:
Using the chain rule:
Thus, we have:
Next, for the partial derivative with respect to
:
Similarly:
Thus:
In summary, the first partial derivatives are:
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