Question

question
determine the derivative of 6^x log_3(x).
select the correct answer below:
\frac{6^x}{ln(3)}+x(log_3(x))(6^x)
\frac{6^x}{xln(3)}+(log_3(x))(6^x)
\frac{6^x}{ln(3)}+x(log_3(x))(6^x)(ln(6))
\frac{6^x}{xln(3)}+(log_3(x))(6^x)(ln(6))

Explanation:

Step1: Apply product - rule

$(uv)' = u'v+uv'$, where $u = 6^{x}$ and $v=\log_{3}(x)$.

Step2: Find $u'$

The derivative of $a^{x}$ is $a^{x}\ln(a)$, so $u'=6^{x}\ln(6)$.

Step3: Find $v'$

The derivative of $\log_{a}(x)$ is $\frac{1}{x\ln(a)}$, so $v'=\frac{1}{x\ln(3)}$.

Step4: Calculate $(uv)'$

$(uv)'=6^{x}\ln(6)\log_{3}(x)+\frac{6^{x}}{x\ln(3)}$.

Answer:

$\frac{6^{x}}{x\ln(3)}+(\log_{3}(x))(6^{x})(\ln(6))$