Find y' for y = e^(10^x)

Find $y'$ for $y = e^{10^{x}}$.

Step 1: Set up the two layers

Let $u = 10^{x}$, so that $y = e^{u}$ and, by the rule of Problem 18, \[\frac{dy}{du} = e^{u} = e^{10^{x}}, \qquad \frac{du}{dx} = 10^{x}\ln 10.\]

Step 2: Multiply

\[ y' = e^{10^{x}}\cdot 10^{x}\ln 10. \]

Step 3: Verification

\[x = 0:\quad 10^{0} = 1 \;\Longrightarrow\; y = e,\quad y' = 1\cdot\ln 10\cdot e \approx 2.30259\cdot 2.71828 \approx 6.25902\]

\[\frac{y(10^{-6})-y(-10^{-6})}{2\times 10^{-6}} \approx 6.25902 \;\;\checkmark\]