DIFFERENTIATION USING THE PRODUCT RULE
The following problem requires the use of the product rule. In the following discussion and solution the derivative of a function h(x) will be denoted by or h'(x) . The product rule is a formal rule for differentiating problems where one function is multiplied by another. The rule follows from the limit definition of derivative and is given by
D{f(x)g(x)} = f(x)g^'(x) + f^'(x)g(x)\]
Differentiate

\[y = x^{3} lnx\]

Solution:
\[ y^{'} = x^3D(lnx) + D(x^{3})lnx\]
\[ = \frac{x^3}{x} + 3x^{2}lnx\]
\[= x^2 + 3x^{2}lnx\]
\[ = x^{2}(1 + 3lnx)\]