A2.8

verify, using technology (e.g., calculator, graphing technology), that the derivative of the exponential function $f(x)=a^x$ is $f^{\prime }(x)=a^x\ln a$ for various values of $a$ [e.g., verifying numerically for $f(x)=2^x$ that $f^{\prime }(x)=2^x\ln 2$ by using a calculator to show that ${\lim }_{h\to 0}\frac{(2^h-1)}{h}$ is $\ln 2$ or by graphing $f(x)=2^x$, determining the value of the slope and the value of the function for specific x-values, and comparing the ratio $\frac{f^{\prime }(x)}{f(x)}$ with $\ln 2$] Sample problem: Given $f(x)=e^x$, verify numerically with technology using ${\lim }_{h\to 0}\frac{(e^{x+h}-e^x)}{h}$ that $f^{\prime }(x)=f(x)\ln e$.