Questions: We obtained that f'(g(x))=4 sec^2(5^x) and g'(x)=5^x ln 5. Apply the Chain Rule u'(x)=f'(g(x)) * g'(x) to find the derivative of u.

We obtained that f'(g(x))=4 sec^2(5^x) and g'(x)=5^x ln 5. Apply the Chain Rule u'(x)=f'(g(x)) * g'(x) to find the derivative of u.
Transcript text: We obtained that $\mathrm{f}^{\prime}(\mathrm{g}(\mathrm{x}))=4 \sec ^{2}\left(5^{\mathrm{x}}\right)$ and $\mathrm{g}^{\prime}(\mathrm{x})=5^{\mathrm{x}} \ln 5$. Apply the Chain Rule $\mathrm{u}^{\prime}(\mathrm{x})=\mathrm{f}^{\prime}(\mathrm{g}(\mathrm{x})) \cdot \mathrm{g}^{\prime}(\mathrm{x})$ to find the derivative of u.
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Solution

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Solution Steps

Step 1: Recall the Chain Rule

The chain rule states that if $u(x) = f(g(x))$, then $u'(x) = f'(g(x)) \cdot g'(x)$.

Step 2: Substitute the given values

We are given $f'(g(x)) = 4\sec^2(5^x)$ and $g'(x) = 5^x\ln 5$. Substituting these into the chain rule formula gives us: $u'(x) = (4\sec^2(5^x)) \cdot (5^x\ln 5)$

Step 3: Simplify

$u'(x) = 4\ln 5 \cdot 5^x\sec^2(5^x)$

Final Answer

$u'(x) = 4\ln 5 \cdot 5^x\sec^2(5^x)$

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