Differentiation enables the calculation of the instantaneous rate of change of functions. In particular, the product and quotient rules allow us to tackle functions expressed as the product or quotient of two other functions.
Introduction
In various situations, functions are intertwined through operations of multiplication or division. Understanding and applying the product and quotient rules is crucial for differentiating such compound expressions.
Product and Quotient Rules
Product Rule:
When two functions u and v are multiplied together:
dxd(uv)=udxdv+vdxduExample:
Differentiate x2ln(x).
Solution:
u=x2,dxdu=2xv=ln(x),dxdv=x1dxd(x2ln(x))=x2⋅x1+ln(x)⋅2x=x+2xln(x)Quotient Rule:
When a function u is divided by another function v:
dxd(vu)=v2vdxdu−udxdvExample:
Differentiate e1−x2x.
Solution:
u=x,dxdu=1v=e1−x2,dxdv=−2xe1−x2
dxd(e1−x2x)=(e1−x2)2e1−x2⋅1−x⋅(−2xe1−x2)=ex2−11+2x2