Question

If \(y = 3^x + e^x + x^x + x^3\), then \(\frac{dy}{dx}\) at \(x=3\) is equal to

• A. $e^3 + 27\log_e 3 + 54$
• B. $e^3 + 27\log_e 3 + 27$
• C. $e^3 + 54\log_e 3 + 27$
• D. $e^3 + 54\log_e 3 + 54$
• E. $e^3 + 54\log_e 3 + 54$

Hint:

Remember $x^x$ derivative: $x^x(1+\ln x)$ — very important!

Answer 1

The Correct Option is D

Concept: Differentiate term-wise: \[ \frac{d}{dx}(a^x)=a^x\ln a,\quad \frac{d}{dx}(x^x)=x^x(1+\ln x) \]

Step 1: Differentiate each term
\[ \frac{dy}{dx} = 3^x\ln3 + e^x + x^x(1+\ln x) + 3x^2 \]

Step 2: Substitute $x=3$
\[ = 27\ln3 + e^3 + 27(1+\ln3) + 27 \]

Step 3: Simplify
\[ = e^3 + 27\ln3 + 27 + 27\ln3 + 27 \] \[ = e^3 + 54\ln3 + 54 \] Final Conclusion:
Option (D)