Question

attempt 1: 10 attempts remaining. use the rules of derivatives to calculate the derivative of the following function and simplify if possible. $f(x)=5ln(x)+7$ $g(x)=$

Explanation:

Step1: Recall derivative rules

The derivative of a sum of functions is the sum of their derivatives, i.e., if $y = u + v$, then $y'=u'+v'$. Also, the derivative of a constant $C$ is $0$ ($(C)' = 0$), and the derivative of $\ln(x)$ is $\frac{1}{x}$, and for a constant - multiple function $y = a\cdot f(x)$ where $a$ is a constant, $y'=a\cdot f'(x)$.

Step2: Differentiate each term

Let $u = 5\ln(x)$ and $v = 7$.
For $u = 5\ln(x)$, using the constant - multiple rule and the derivative of $\ln(x)$, we have $u'=5\times\frac{1}{x}=\frac{5}{x}$.
For $v = 7$, since it is a constant, $v' = 0$.

Step3: Find the derivative of the function

By the sum - rule of derivatives, $f'(x)=u'+v'$. Substituting the values of $u'$ and $v'$, we get $f'(x)=\frac{5}{x}+0=\frac{5}{x}$.

Answer:

$\frac{5}{x}$