Question

1. differentiate y=(x² - 5)lnx answer: $\frac{dy}{dx}=$

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$. Let $u = x^{2}-5$ and $v=\ln x$.

Step2: Differentiate $u$ and $v$

Differentiate $u = x^{2}-5$ with respect to $x$: $\frac{du}{dx}=2x$. Differentiate $v=\ln x$ with respect to $x$: $\frac{dv}{dx}=\frac{1}{x}$.

Step3: Substitute into product - rule

$\frac{dy}{dx}=(x^{2}-5)\cdot\frac{1}{x}+\ln x\cdot(2x)$.

Step4: Simplify the expression

$\frac{dy}{dx}=\frac{x^{2}-5}{x}+2x\ln x=x - \frac{5}{x}+2x\ln x$.

Answer:

$x-\frac{5}{x}+2x\ln x$