How do you derive the substitution effect and income effect using Slutsky equation?

The Slutsky equation breaks down the total effect of a price change on the quantity demanded into two components: the substitution effect and the income effect. The substitution effect reflects how a change in the price of a good alters its relative attractiveness compared to other goods, leading to a change in consumption while keeping utility constant. The income effect, on the other hand, captures how a price change affects the consumer's purchasing power, thus altering the quantity demanded based on the new utility-maximizing consumption bundle. Mathematically, the Slutsky equation is expressed as ( \frac{\partial x}{\partial p} = \frac{\partial h}{\partial p} - h \frac{\partial x}{\partial I} ), where ( \frac{\partial x}{\partial p} ) is the total effect, ( \frac{\partial h}{\partial p} ) is the substitution effect, and ( -h \frac{\partial x}{\partial I} ) is the income effect.