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Butoxors [25]
2 years ago
8

Ya'll fr get 50 points if you answer this. What is the rate of change of

" id="TexFormula1" title="f(x)=2^x" alt="f(x)=2^x" align="absmiddle" class="latex-formula">?
Mathematics
1 answer:
Evgesh-ka [11]2 years ago
8 0

Answer:

\dfrac{dy}{dx}=2^x\ln 2

Step-by-step explanation:

**This is a non-linear function and therefore <u>does not have a constant rate of change</u>.  It will have a different slope depending on what points you use in the average rate of change formula:\mathsf{average \ rate \ of \ change = \dfrac{change \ in \ y}{change \ in \ x}}

To calculate rate of change, differentiate.

substitute y for f(x):  

\implies y=2^x

Take natural logs of both sides:

\implies \ln y=\ln 2^x

Apply the log rule  \ln a^b=b \ln a :

\implies \ln y=x\ln 2

Differentiate with respect to x:

\implies \dfrac{1}{y} \frac{dy}{dx}=\ln 2

Mulitply both sides by y:

\implies \dfrac{dy}{dx}=y\ln 2

Replace y with y=2^x

\implies \dfrac{dy}{dx}=2^x\ln 2

Therefore, rate of change of the function is :

\dfrac{dy}{dx}=2^x\ln 2

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