Question

find f(x). f(x)=6x - 5ln x^{2} f(x)=square

Explanation:

Step1: Differentiate 6x

The derivative of $ax$ with respect to $x$ is $a$. So the derivative of $6x$ with respect to $x$ is 6, i.e., $\frac{d}{dx}(6x)=6$.

Step2: Differentiate $- 5\ln x^{2}$

First, use the property $\ln x^{n}=n\ln x$. So $-5\ln x^{2}=- 10\ln x$. The derivative of $\ln x$ is $\frac{1}{x}$, and by the constant - multiple rule $\frac{d}{dx}(cf(x)) = c\frac{d}{dx}(f(x))$. Then $\frac{d}{dx}(-10\ln x)=-10\times\frac{1}{x}=-\frac{10}{x}$.

Step3: Combine the derivatives

By the sum - difference rule of differentiation $\frac{d}{dx}(u(x)\pm v(x))=\frac{d}{dx}(u(x))\pm\frac{d}{dx}(v(x))$. Here $u(x) = 6x$ and $v(x)=5\ln x^{2}$, so $f'(x)=\frac{d}{dx}(6x)-\frac{d}{dx}(5\ln x^{2})=6-\frac{10}{x}$.

Answer:

$6-\frac{10}{x}$