Differentiate wrt x
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Let then
and then
To differentiate the function with respect to , you can use the product rule. The product rule states that if you have a function of the form , the derivative is given by:
.
In this case, let and .
Now, we’ll find their derivatives:
(using the power rule).
To find , we’ll need to use the chain rule. The chain rule states that if you have a composite function, such as , the derivative of with respect to is given by:
.
Let , so .
Now, we can find :
.
(the derivative of is , and the derivative of a constant, , is ).
So, .
Now, apply the product rule:
.
Now, simplify this expression:
.
So, the derivative of with respect to is:
.