Question

If \( f(x) = x^6 + 6^x \), then \( f'(x) \) is equal to

• A. \( 12x \)
• B. \( x+4 \)
• C. \( 6x^5 + 6^x \log 6 \)
• D. \( 6x^5 + x6^{x-1} \)
• E. \( x^6 \)

Hint:

Remember derivative of \( a^x \) includes \( \ln a \).

Answer 1

The Correct Option is C

Concept: Derivative rules: \[ \frac{d}{dx}(x^n) = nx^{n-1}, \quad \frac{d}{dx}(a^x) = a^x \ln a \]

Step 1: Differentiate \( x^6 \).
\[ = 6x^5 \]

Step 2: Differentiate \( 6^x \).
\[ = 6^x \ln 6 \]

Step 3: Combine derivatives.
\[ f'(x) = 6x^5 + 6^x \ln 6 \]

Step 4: Verify no simplification needed.
Expression is already simplest form.

Step 5: Final answer.
\[ 6x^5 + 6^x \log 6 \]