Question

attempt 1: 10 attempts remaining. find the derivative ( y ) of the function ( y = 5x - x ln(x) ). ( y = ) input box submit answer next item

Explanation:

Step1: Differentiate term by term

The function is \( y = 5x - x\ln(x) \). We use the sum/difference rule of differentiation, which states that \( (u - v)' = u' - v' \). So we differentiate \( 5x \) and \( x\ln(x) \) separately.

For the first term, \( \frac{d}{dx}(5x) \): Using the power rule \( \frac{d}{dx}(ax^n)=nax^{n - 1} \), here \( a = 5 \) and \( n = 1 \), so \( \frac{d}{dx}(5x)=5\times1\times x^{1 - 1}=5 \).

For the second term, \( \frac{d}{dx}(x\ln(x)) \): We use the product rule \( (uv)' = u'v+uv' \), where \( u = x \) and \( v=\ln(x) \).

• First, find \( u' \): \( \frac{d}{dx}(x)=1 \) (by power rule, \( n = 1 \), \( a = 1 \), so \( 1\times x^{0}=1 \)).
• Second, find \( v' \): \( \frac{d}{dx}(\ln(x))=\frac{1}{x} \).

Then by product rule, \( (x\ln(x))'=u'v + uv'=1\times\ln(x)+x\times\frac{1}{x}=\ln(x) + 1 \).

Step2: Combine the derivatives

Now, using the difference rule: \( y'=\frac{d}{dx}(5x)-\frac{d}{dx}(x\ln(x)) \). Substituting the derivatives we found: \( y' = 5-(\ln(x)+1) \).

Simplify the expression: \( y'=5 - \ln(x)-1=4-\ln(x) \).

Answer:

\( 4 - \ln(x) \)