find f(x).\nf(x)= - 6\\ln x + 5x^{2}-10\nf(x)=\\square

Explanation:

Step1: Differentiate -6ln(x)

The derivative of $\ln(x)$ is $\frac{1}{x}$. Using the constant - multiple rule, the derivative of $- 6\ln(x)$ is $-6\times\frac{1}{x}=-\frac{6}{x}$.

Step2: Differentiate $5x^{2}$

Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, for $n = 2$, the derivative of $5x^{2}$ is $5\times2x=10x$.

Step3: Differentiate the constant

The derivative of a constant $C$ (here $C=-10$) is 0.

Step4: Combine the derivatives

By the sum - difference rule of differentiation, $f'(x)$ is the sum of the derivatives of each term. So $f'(x)=-\frac{6}{x}+10x + 0$.

Answer:

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