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What is the derivative of #b^x# where b is a constant?

Answer 1

Emilia Aguilar

#d/dx[b^x]=b^x*lnb#

Initially, take note of the fact that

#b^x=e^ln(b^x)=e^(xlnb)#

This enables us to apply the chain rule to distinguish the function:

#d/dx[e^(xlnb)]=e^(xlnb)*d/dx[xlnb]#

Just like #d/dx[5x]=5#, #d/dx[xlnb]=lnb#, since #lnb# will always be a constant.

We now have a derivative of:

#e^(xlnb)*lnb#

Now, recall that #e^(xlnb)=b^x#. This gives us our final, differentiated result:

#d/dx[b^x]=b^x*lnb#

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Answer 2

Aurora Alvarado

The derivative of ( b^x ) where ( b ) is a constant is ( b^x \cdot \ln(b) ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.