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Questions: Algebra BusinessCalculus
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Find the derivative of \(\displaystyle f(x) = e^{e^{e^{x}}}\)
Taking the derivative with the chain rule gives \(\displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx}\). \(\displaystyle f'(x) = (e^{u})(e^{v})(e^{x})\). Substituting back in \(\displaystyle u\) and \(\displaystyle v\) gives \(\displaystyle f'(x) = e^{v} e^{x} e^{e^{v}} = e^{x} e^{e^{x}} e^{e^{e^{x}}}\).
\begin{question}Find the derivative of $f(x) = e^{e^{e^{x}}}$
\soln{9cm}{Taking the derivative with the chain rule gives $f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx}$. $f'(x) = (e^{u})(e^{v})(e^{x})$. Substituting back in $u$ and $v$ gives $f'(x) = e^{v} e^{x} e^{e^{v}} = e^{x} e^{e^{x}} e^{e^{e^{x}}}$. }
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Find the derivative of <img class="equation_image" title=" \displaystyle f(x) = e^{e^{e^{x}}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20e%5E%7Be%5E%7Be%5E%7Bx%7D%7D%7D%20" alt="LaTeX: \displaystyle f(x) = e^{e^{e^{x}}} " data-equation-content=" \displaystyle f(x) = e^{e^{e^{x}}} " /> </p> </p><p> <p>Taking the derivative with the chain rule gives <img class="equation_image" title=" \displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx} " src="/equation_images/%20%5Cdisplaystyle%20f%27%28x%29%20%3D%20%5Cfrac%7Bdf%7D%7Bdu%7D%5Cfrac%7Bdu%7D%7Bdv%7D%5Cfrac%7Bdv%7D%7Bdx%7D%20" alt="LaTeX: \displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx} " data-equation-content=" \displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx} " /> . <img class="equation_image" title=" \displaystyle f'(x) = (e^{u})(e^{v})(e^{x}) " src="/equation_images/%20%5Cdisplaystyle%20f%27%28x%29%20%3D%20%28e%5E%7Bu%7D%29%28e%5E%7Bv%7D%29%28e%5E%7Bx%7D%29%20" alt="LaTeX: \displaystyle f'(x) = (e^{u})(e^{v})(e^{x}) " data-equation-content=" \displaystyle f'(x) = (e^{u})(e^{v})(e^{x}) " /> . Substituting back in <img class="equation_image" title=" \displaystyle u " src="/equation_images/%20%5Cdisplaystyle%20u%20" alt="LaTeX: \displaystyle u " data-equation-content=" \displaystyle u " /> and <img class="equation_image" title=" \displaystyle v " src="/equation_images/%20%5Cdisplaystyle%20v%20" alt="LaTeX: \displaystyle v " data-equation-content=" \displaystyle v " /> gives <img class="equation_image" title=" \displaystyle f'(x) = e^{v} e^{x} e^{e^{v}} = e^{x} e^{e^{x}} e^{e^{e^{x}}} " src="/equation_images/%20%5Cdisplaystyle%20f%27%28x%29%20%3D%20e%5E%7Bv%7D%20e%5E%7Bx%7D%20e%5E%7Be%5E%7Bv%7D%7D%20%3D%20e%5E%7Bx%7D%20e%5E%7Be%5E%7Bx%7D%7D%20e%5E%7Be%5E%7Be%5E%7Bx%7D%7D%7D%20" alt="LaTeX: \displaystyle f'(x) = e^{v} e^{x} e^{e^{v}} = e^{x} e^{e^{x}} e^{e^{e^{x}}} " data-equation-content=" \displaystyle f'(x) = e^{v} e^{x} e^{e^{v}} = e^{x} e^{e^{x}} e^{e^{e^{x}}} " /> . </p> </p>