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So in math applications, we often use exponential functions
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, trigger trigger metric functions. So for that reason
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, it's helpful for us to be able to,
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um, essentially take the derivative of those when wearing
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calculus, the derivatives, such a crucial part of
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understanding rates of change and all those things that we
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want to be able to use logarithms exponents, trigger
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metric functions and be able to take their derivatives.
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So one way in which we do this is if
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we have a complicated function such as this one right
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here where we have, um, the exponential function
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, but the variables in both the base and the
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exponents, so only that we solve this is taking
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the natural log of both sides. When we do
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that, this coastline X is now power so it
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can move out in front and multiply. Then the
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natural log of why Now we can perform implicit differentiation
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. So what we'll have is the the D.
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X of the natural log of y is going to
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be won over. Why Times y prime and then
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uh co sign of X Natural times Natural. Lagerback's
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the derivative that will be co sign of X times
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one over X class Attrill log of acts times a
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negative sine x So when we combine, um,
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these right here we'll get the coastline of X over
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X. So let's because I an X over X
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Then we can get rid of all this. Um
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, and then what we'll have is the negative sign
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X times the natural log of X. So that
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will be a negative natural log of X times sine
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x. And then lastly, we want to multiply
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by. Why on both sides? Yeah, so
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out here will be why, but we know that
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why is X to the coastline X So that's we
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have our final answer. Um, like this.
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And we can even move the cynics depending on where
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we want to write it. But it ultimately means
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the same thing that's right here will be our final
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answer. Using implicit differentiation. Um, derivatives of
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logarithms s so on and so forth