Exercise · ZW-DERIVATIVES-OF-EXPONENTIALS-AND-LOGARITHMS-007 · ★★☆ Intermediate
Differentiate \(f(x)=e^{x}\ln x\) for \(x>0\).
Choose one option.
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- D. \(e^{x}\ln x+\frac{e^{x}}{x}\)
Explanation
By the product rule: \((e^{x})'\ln x+e^{x}(\ln x)'=e^{x}\ln x+\frac{e^{x}}{x}\).
Review the concept: Derivatives of exponentials and logarithms
The function \(e^x\) is its own derivative. For base \(a>0\), \(a^x\) has derivative \(a^x\ln a\), and \(\ln x\) has derivative \(\frac{1}{x}\) for \(x>0\).
\[(e^{x})'=e^{x},\quad(a^{x})'=a^{x}\ln a,\quad(\ln x)'=\frac{1}{x}\]
More practice problems: derivatives of exponentials and logarithms
- ZW-DERIVATIVES-OF-EXPONENTIALS-AND-LOGARITHMS-001 Differentiate f(x)=e^3 x.
- ZW-DERIVATIVES-OF-EXPONENTIALS-AND-LOGARITHMS-002 Differentiate f(x)=2^x.
- ZW-DERIVATIVES-OF-EXPONENTIALS-AND-LOGARITHMS-003 Differentiate f(x)=ln (3 x ).
- ZW-DERIVATIVES-OF-EXPONENTIALS-AND-LOGARITHMS-004 Differentiate f(x)=5^x.
- ZW-DERIVATIVES-OF-EXPONENTIALS-AND-LOGARITHMS-005 Differentiate f(x)=e^-x.
- ZW-DERIVATIVES-OF-EXPONENTIALS-AND-LOGARITHMS-006 Differentiate f(x)=ln x for x>0.
- ZW-DERIVATIVES-OF-EXPONENTIALS-AND-LOGARITHMS-008 Differentiate f(x)=log_3(x^2+1).
- ZW-DERIVATIVES-OF-EXPONENTIALS-AND-LOGARITHMS-009 Compute (e^x)' at x=0.
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