What is the result of differentiating (x^4/4) ln(x) - (x^4/16) + C?

To find the result of differentiating the expression ( \frac{x^4}{4} \ln(x) - \frac{x^4}{16} + C ), we can apply the product rule and the basic rules of differentiation.

Start with the term ( \frac{x^4}{4} \ln(x) ). This is a product of two functions: ( u = \frac{x^4}{4} ) and ( v = \ln(x) ).

Using the product rule, which states that ( (uv)' = u'v + uv' ), we will determine ( u' ) and ( v' ):

1. ( u' = \frac{d}{dx}\left(\frac{x^4}{4}\right) = x^3 ).

2. ( v' = \frac{d}{dx}(\ln(x)) = \frac{1}{x} ).

Applying the product rule:

[

\frac{d}{dx}\left(\frac{x^4}{4} \ln(x)\right) = \left(x^3\right)\ln(x) + \left(\frac{x^4}{4}\cdot\frac{1}{x