Question
If \( f(x)=7 e^{x}+x^{e} \), find \( f^{\prime}(x) \). \( \square \)
Ask by Elliott Savage. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The derivative of \( f(x) \) is \( f'(x) = 7e^{x} + e\,x^{e-1} \).
Solution
1. Write the function:
\[
f(x)=7e^{x}+x^{e}.
\]
2. Differentiate the function term-by-term.
3. The derivative of the first term is:
\[
\frac{d}{dx}\left(7e^{x}\right)=7e^{x}.
\]
4. The derivative of the second term, using the power rule for \( x^n \) where \( n \) is a constant, is:
\[
\frac{d}{dx}\left(x^{e}\right)=e\,x^{e-1}.
\]
5. Therefore, the derivative of \( f(x) \) is:
\[
f'(x)=7e^{x}+e\,x^{e-1}.
\]
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Simplify this solution Mind Expander
To find the derivative of the function \( f(x)=7 e^{x}+x^{e} \), we can use the rules of differentiation. The derivative of \( e^x \) is \( e^x \), and the derivative of \( x^n \) is \( n x^{n-1} \). Thus, we calculate: 1. The derivative of \( 7 e^x \) is \( 7 e^x \). 2. The derivative of \( x^{e} \) is \( e x^{e-1} \). Combining these results, we get the derivative: \[ f^{\prime}(x) = 7 e^x + e x^{e-1}. \]
