# Math 22 Differentials and Marginal Analysis

## Differentials

 Let ${\displaystyle y=f(x)}$ represent a differentiable function. The differential of ${\displaystyle x}$ (denoted by ${\displaystyle dx}$)
is any nonzero real number. The differential of ${\displaystyle y}$ (denoted by ) is ${\displaystyle dy=f'(x)dx}$.


Example: 1) Consider the function ${\displaystyle f(x)=3x^{3}}$. Find ${\displaystyle dy}$ when ${\displaystyle x=1}$ and ${\displaystyle dx=0.01}$

Solution:
Notice: ${\displaystyle f(x)=3x^{3}}$, so ${\displaystyle dy=f'(x)dx=9x^{2}dx=9(1)^{2}.(0.01)=0.09}$

2) Find ${\displaystyle dy}$ of each function below:

a) ${\displaystyle y={\frac {5x+7}{15}}}$

Solution:
Notice: ${\displaystyle y={\frac {5x+7}{15}}={\frac {5x}{15}}+{\frac {7}{15}}}$, so
${\displaystyle dy=f'(x)dx={\frac {5}{15}}={\frac {1}{3}}dx}$

b) ${\displaystyle y=x(1.25+0.02{\sqrt {x}})}$

Solution:
Notice: ${\displaystyle y=x(1.25+0.02{\sqrt {x}})=1.25x+0.02x{\sqrt {x}}=1.25x+0.02x^{\frac {3}{2}}}$, so ${\displaystyle dy=f'(x)dx=[1.25+0.02({\frac {3}{2}})x^{\frac {1}{2}}]dx=[1.25+0.03{\sqrt {x}}]dx}$