Question

attempt 1: 10 attempts remaining. differentiate $y = (x^2 - 5)ln x$ answer: $\frac{dy}{dx} = $ submit answer next item

Explanation:

Step1: Identify the product rule

We have \( y=(x^{2}-5)\ln x \), which is a product of two functions \( u = x^{2}-5 \) and \( v=\ln x \). The product rule for differentiation is \( \frac{d}{dx}(uv)=u'v + uv' \).

Step2: Differentiate \( u \) and \( v \)

• Differentiate \( u = x^{2}-5 \) with respect to \( x \): \( u'=\frac{d}{dx}(x^{2}-5)=2x \)
• Differentiate \( v=\ln x \) with respect to \( x \): \( v'=\frac{d}{dx}(\ln x)=\frac{1}{x} \)

Step3: Apply the product rule

Using the product rule \( \frac{dy}{dx}=u'v + uv' \), substitute \( u = x^{2}-5 \), \( u' = 2x \), \( v=\ln x \), and \( v'=\frac{1}{x} \):
\[

$$\begin{align*} \frac{dy}{dx}&=2x\cdot\ln x+(x^{2}-5)\cdot\frac{1}{x}\\ &=2x\ln x+\frac{x^{2}-5}{x}\\ &=2x\ln x + x-\frac{5}{x} \end{align*}$$

\]

Answer:

\( 2x\ln x + x-\frac{5}{x} \)