Question

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What is the rate of change of $f(x)=2^x$?

Answer 1

Answer:

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

Step-by-step explanation:

**This is a non-linear function and therefore does not have a constant rate of change.  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 :

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