find the indicated derivative and simplify.\n$\frac{d}{dx}2xlog x^{6}$\n$\frac{d}{dx}2xlog…

find the indicated derivative and simplify.\n$\frac{d}{dx}2xlog x^{6}$\n$\frac{d}{dx}2xlog x^{6}=square$\n(type an exact answer.)

find the indicated derivative and simplify.\n$\frac{d}{dx}2xlog x^{6}$\n$\frac{d}{dx}2xlog x^{6}=square$\n(type an exact answer.)

Answer

Explanation:

Step1: Use logarithm property

First, use the power - rule of logarithms $\log x^{n}=n\log x$. So, $2x\log x^{6}=2x\times6\log x = 12x\log x$.

Step2: Apply the product - rule

The product - rule states that if $y = uv$, where $u$ and $v$ are functions of $x$, then $y^\prime=u^\prime v + uv^\prime$. Let $u = 12x$ and $v=\log x$. The derivative of $u$ with respect to $x$ is $u^\prime=\frac{d}{dx}(12x)=12$, and the derivative of $v$ with respect to $x$ is $v^\prime=\frac{d}{dx}(\log x)=\frac{1}{x\ln 10}$ (assuming the common logarithm $\log=\log_{10}$).

Step3: Calculate the derivative

By the product - rule, $\frac{d}{dx}(12x\log x)=12\log x+12x\times\frac{1}{x\ln 10}$.

Step4: Simplify the result

$12\log x + \frac{12}{\ln 10}$.

Answer:

$12\log x+\frac{12}{\ln 10}$