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Calculus
Derivatives
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Find the derivative of \(\displaystyle f(x) = 8^{8^{8^{x}}}\)

Taking the derivative with the chain rule gives \(\displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx}\). \(\displaystyle f'(x) = (8^{u} \ln{\left(8 \right)})(8^{v} \ln{\left(8 \right)})(8^{x} \ln{\left(8 \right)})\). Substituting back in \(\displaystyle u\) and \(\displaystyle v\) gives \(\displaystyle f'(x) = 8^{8^{v}} 8^{v} 8^{x} \ln{\left(8 \right)}^{3} = 8^{8^{8^{x}}} 8^{8^{x}} 8^{x} \ln{\left(8 \right)}^{3}\).

Download \(\LaTeX\) 

\begin{question}Find the derivative of $f(x) = 8^{8^{8^{x}}}$
    \soln{9cm}{Taking the derivative with the chain rule gives $f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx}$. $f'(x) = (8^{u} \ln{\left(8 \right)})(8^{v} \ln{\left(8 \right)})(8^{x} \ln{\left(8 \right)})$. Substituting back in $u$ and $v$ gives $f'(x) = 8^{8^{v}} 8^{v} 8^{x} \ln{\left(8 \right)}^{3} = 8^{8^{8^{x}}} 8^{8^{x}} 8^{x} \ln{\left(8 \right)}^{3}$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle f(x) = 8^{8^{8^{x}}} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%208%5E%7B8%5E%7B8%5E%7Bx%7D%7D%7D%20" alt="LaTeX:  \displaystyle f(x) = 8^{8^{8^{x}}} " data-equation-content=" \displaystyle f(x) = 8^{8^{8^{x}}} " /> </p> </p>
HTML for Canvas
<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) = (8^{u} \ln{\left(8 \right)})(8^{v} \ln{\left(8 \right)})(8^{x} \ln{\left(8 \right)}) " src="/equation_images/%20%5Cdisplaystyle%20f%27%28x%29%20%3D%20%288%5E%7Bu%7D%20%5Cln%7B%5Cleft%288%20%5Cright%29%7D%29%288%5E%7Bv%7D%20%5Cln%7B%5Cleft%288%20%5Cright%29%7D%29%288%5E%7Bx%7D%20%5Cln%7B%5Cleft%288%20%5Cright%29%7D%29%20" alt="LaTeX:  \displaystyle f'(x) = (8^{u} \ln{\left(8 \right)})(8^{v} \ln{\left(8 \right)})(8^{x} \ln{\left(8 \right)}) " data-equation-content=" \displaystyle f'(x) = (8^{u} \ln{\left(8 \right)})(8^{v} \ln{\left(8 \right)})(8^{x} \ln{\left(8 \right)}) " /> . 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) = 8^{8^{v}} 8^{v} 8^{x} \ln{\left(8 \right)}^{3} = 8^{8^{8^{x}}} 8^{8^{x}} 8^{x} \ln{\left(8 \right)}^{3} " src="/equation_images/%20%5Cdisplaystyle%20f%27%28x%29%20%3D%208%5E%7B8%5E%7Bv%7D%7D%208%5E%7Bv%7D%208%5E%7Bx%7D%20%5Cln%7B%5Cleft%288%20%5Cright%29%7D%5E%7B3%7D%20%3D%208%5E%7B8%5E%7B8%5E%7Bx%7D%7D%7D%208%5E%7B8%5E%7Bx%7D%7D%208%5E%7Bx%7D%20%5Cln%7B%5Cleft%288%20%5Cright%29%7D%5E%7B3%7D%20" alt="LaTeX:  \displaystyle f'(x) = 8^{8^{v}} 8^{v} 8^{x} \ln{\left(8 \right)}^{3} = 8^{8^{8^{x}}} 8^{8^{x}} 8^{x} \ln{\left(8 \right)}^{3} " data-equation-content=" \displaystyle f'(x) = 8^{8^{v}} 8^{v} 8^{x} \ln{\left(8 \right)}^{3} = 8^{8^{8^{x}}} 8^{8^{x}} 8^{x} \ln{\left(8 \right)}^{3} " /> . </p> </p>