Questions: Find the following derivative. d/dx(5x^5 ln x)

Solution Steps

We start with the function \( f(x) = 5x^5 \ln x \).

Using the product rule, we identify \( u(x) = 5x^5 \) and \( v(x) = \ln x \). The product rule states: \[ \frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v' \]

We calculate the derivatives:

• \( u' = \frac{d}{dx}(5x^5) = 25x^4 \)
• \( v' = \frac{d}{dx}(\ln x) = \frac{1}{x} \)

Substituting \( u \), \( u' \), \( v \), and \( v' \) into the product rule gives: \[ \frac{d}{dx}(5x^5 \ln x) = 25x^4 \ln x + 5x^5 \cdot \frac{1}{x} \]

Simplifying the second term: \[ 5x^5 \cdot \frac{1}{x} = 5x^4 \] Thus, the derivative simplifies to: \[ \frac{d}{dx}(5x^5 \ln x) = 25x^4 \ln x + 5x^4 \]

Final Answer