Question

find f(x).
f(x)= - 7\ln x + 5x^{2}-7
f(x)=\square

Explanation:

Step1: Recall derivative rules

Use the sum - difference rule of derivatives $\frac{d}{dx}(u\pm v\pm w)=\frac{du}{dx}\pm\frac{dv}{dx}\pm\frac{dw}{dx}$, and the derivative formulas $\frac{d}{dx}(\ln x)=\frac{1}{x}$, $\frac{d}{dx}(ax^{n}) = nax^{n - 1}$, $\frac{d}{dx}(c)=0$ where $a,c$ are constants.

Step2: Differentiate each term

For $y=-7\ln x$, $\frac{d}{dx}(-7\ln x)=-7\times\frac{1}{x}=-\frac{7}{x}$; for $y = 5x^{2}$, $\frac{d}{dx}(5x^{2})=2\times5x=10x$; for $y=-7$, $\frac{d}{dx}(-7)=0$.

Step3: Combine the results

$f^{\prime}(x)=\frac{d}{dx}(-7\ln x)+\frac{d}{dx}(5x^{2})+\frac{d}{dx}(-7)=-\frac{7}{x}+10x + 0$.

Answer:

$-\frac{7}{x}+10x$