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What is the derivative of #f(x)=log_4(e^x+3)# ?

Answer 1

First, we will rewrite the function in terms of natural logarithms, using the change-of-base rule:

#f(x) = ln(e^x + 3)/ln4#

Differentiating will require use of the chain rule:

#d/dx f(x) = 1/ln 4 * d/(d(e^x + 3))[ln(e^x + 3)] * d/dx[e^x + 3]#

We know that since the derivative of #ln x# with respect to #x# is #1/x#, then the derivative of #ln(e^x + 3)# with respect to #e^x + 3# will be #1/(e^x + 3)#. We also know that the derivative of #e^x + 3# with respect to #x# will simply be #e^x#:

#d/dx f(x) = 1/ln 4 * 1/(e^x + 3) * (e^x)#

Simplifying yields:

#d/dx f(x) = (e^x)/(ln 4(e^x + 3)) #

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Answer 2

Paisley Johnson

To find the derivative of ( f(x) = \log_4(e^x + 3) ), apply the chain rule:

[ f'(x) = \frac{1}{\ln(4)} \cdot \frac{1}{e^x + 3} \cdot e^x ]

So, the derivative of ( f(x) ) is:

[ f'(x) = \frac{e^x}{\ln(4)(e^x + 3)} ]

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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.